Chapter 7: Integrals
Complete Board Exam Focused Notes with Standard Formulas & PYQs
Exam Weightage & Blueprint
Total: ~20 MarksIntegrals is one of the highest weightage chapters in Class 12 Mathematics. It forms the foundation for many practical applications in physics and engineering.
| Question Type | Marks | Frequency | Focus Topic |
|---|---|---|---|
| MCQ | 1 | Very High | Standard Integrals, Integration by Parts |
| Short Answer (2M) | 2 | Very High | Integration by Substitution, Partial Fractions |
| Short Answer (3M) | 3 | High | Definite Integrals, Properties |
| Long Answer (4M) | 4 | Very High | Integration by Parts, Special Functions |
| Long Answer (6M) | 6 | High | Application of Definite Integrals, Complex Problems |
⏰ Last 24-Hour Checklist
Standard Formulas (Must Know!)
- ☐ $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (n ≠ -1)
- ☐ $\int \frac{1}{x} dx = \log|x| + C$
- ☐ $\int e^x dx = e^x + C$
- ☐ $\int a^x dx = \frac{a^x}{\log a} + C$
- ☐ $\int \sin x dx = -\cos x + C$
- ☐ $\int \cos x dx = \sin x + C$
- ☐ $\int \sec^2 x dx = \tan x + C$
- ☐ $\int \text{cosec}^2 x dx = -\cot x + C$
Key Techniques
- ☐ Integration by Substitution
- ☐ Integration by Parts: $\int uv dx = u\int v dx - \int u'(\int v dx)dx$
- ☐ Partial Fractions
- ☐ Standard Forms of Integrals
- ☐ Definite Integral Properties
- ☐ $\int_a^b f(x)dx = F(b) - F(a)$
- ☐ $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$
- ☐ $\int_0^{2a} f(x)dx = 2\int_0^a f(x)dx$ (if f(2a-x)=f(x))
Standard Integrals ★★★★★
Integration is the Inverse Process of Differentiation
If $\frac{d}{dx}F(x) = f(x)$, then $\int f(x)dx = F(x) + C$
Basic Power Functions
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$
$$\int \frac{1}{x} dx = \log|x| + C$$
$$\int dx = x + C$$
Exponential & Logarithmic Functions
$$\int e^x dx = e^x + C$$
$$\int a^x dx = \frac{a^x}{\log a} + C$$
Trigonometric Functions
| Function | Integral |
|---|---|
| $\int \sin x dx$ | $-\cos x + C$ |
| $\int \cos x dx$ | $\sin x + C$ |
| $\int \sec^2 x dx$ | $\tan x + C$ |
| $\int \text{cosec}^2 x dx$ | $-\cot x + C$ |
| $\int \sec x \tan x dx$ | $\sec x + C$ |
| $\int \text{cosec} x \cot x dx$ | $-\text{cosec} x + C$ |
| $\int \tan x dx$ | $\log|\sec x| + C = -\log|\cos x| + C$ |
| $\int \cot x dx$ | $\log|\sin x| + C$ |
| $\int \sec x dx$ | $\log|\sec x + \tan x| + C$ |
| $\int \text{cosec} x dx$ | $\log|\text{cosec} x - \cot x| + C$ |
Inverse Trigonometric Functions
$$\int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1}x + C$$
$$\int \frac{dx}{\sqrt{1-x^2}} = -\cos^{-1}x + C$$
$$\int \frac{dx}{1+x^2} = \tan^{-1}x + C$$
$$\int \frac{dx}{1+x^2} = -\cot^{-1}x + C$$
Memory Aid: ILATE Rule for Integration by Parts
Inverse Trigonometric
Logarithmic
Algebraic
Trigonometric
Exponential
Choose the first function in this order!
Inverse Trigonometric
Logarithmic
Algebraic
Trigonometric
Exponential
Choose the first function in this order!
Integration by Substitution 🔥🔥🔥
Method: When the integrand involves a function and its derivative, substitute the function as a new variable.
If $\int f(x)dx$ is difficult, put $x = g(t)$ so that $dx = g'(t)dt$
If $\int f(x)dx$ is difficult, put $x = g(t)$ so that $dx = g'(t)dt$
Common Substitutions
| Integrand Contains | Substitution |
|---|---|
| $\sqrt{a^2 - x^2}$ | $x = a\sin\theta$ or $x = a\cos\theta$ |
| $\sqrt{a^2 + x^2}$ | $x = a\tan\theta$ or $x = a\cot\theta$ |
| $\sqrt{x^2 - a^2}$ | $x = a\sec\theta$ or $x = a\text{cosec}\theta$ |
| $\sqrt{a-x}$ and $\sqrt{x-a}$ | $x = a + t^2$ or $x = a - t^2$ |
Important Standard Forms
$$\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\log\left|\frac{x-a}{x+a}\right| + C$$
$$\int \frac{dx}{a^2 - x^2} = \frac{1}{2a}\log\left|\frac{a+x}{a-x}\right| + C$$
$$\int \frac{dx}{x^2 + a^2} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C$$
$$\int \frac{dx}{\sqrt{x^2 - a^2}} = \log|x + \sqrt{x^2-a^2}| + C$$
$$\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\frac{x}{a} + C$$
$$\int \frac{dx}{\sqrt{x^2 + a^2}} = \log|x + \sqrt{x^2+a^2}| + C$$
Pro Tip: When you see $ax^2 + bx + c$, always complete the square!
$ax^2 + bx + c = a\left[(x + \frac{b}{2a})^2 + \frac{4ac-b^2}{4a^2}\right]$
$ax^2 + bx + c = a\left[(x + \frac{b}{2a})^2 + \frac{4ac-b^2}{4a^2}\right]$
Integration by Parts ★★★★★
Formula:
$$\int u \cdot v \, dx = u\int v \, dx - \int \left[\frac{du}{dx} \cdot \int v \, dx\right] dx$$
Or simply: $\int uv \, dx = u\int v - \int u'(\int v)$
ILATE Rule (Choose First Function)
1. Inverse Trigonometric ($\sin^{-1}x, \tan^{-1}x$, etc.)
2. Logarithmic ($\log x$)
3. Algebraic ($x, x^2, x^3$, etc.)
4. Trigonometric ($\sin x, \cos x$, etc.)
5. Exponential ($e^x, a^x$)
1. Inverse Trigonometric ($\sin^{-1}x, \tan^{-1}x$, etc.)
2. Logarithmic ($\log x$)
3. Algebraic ($x, x^2, x^3$, etc.)
4. Trigonometric ($\sin x, \cos x$, etc.)
5. Exponential ($e^x, a^x$)
Special Integrals
$$\int e^x[f(x) + f'(x)]dx = e^x f(x) + C$$
Common Examples:
• $\int x\sin x \, dx = -x\cos x + \sin x + C$
• $\int x e^x \, dx = e^x(x-1) + C$
• $\int \log x \, dx = x\log x - x + C$
• $\int x^2 e^x \, dx = e^x(x^2 - 2x + 2) + C$
• $\int x\sin x \, dx = -x\cos x + \sin x + C$
• $\int x e^x \, dx = e^x(x-1) + C$
• $\int \log x \, dx = x\log x - x + C$
• $\int x^2 e^x \, dx = e^x(x^2 - 2x + 2) + C$
Integration by Partial Fractions ★★★★☆
When to Use: When integrand is a rational function $\frac{P(x)}{Q(x)}$ where degree of P(x) < degree of Q(x)
If degree of P(x) ≥ degree of Q(x), first divide P(x) by Q(x)
If degree of P(x) ≥ degree of Q(x), first divide P(x) by Q(x)
Standard Forms
| Form of Q(x) | Partial Fraction |
|---|---|
| $(x-a)(x-b)$ | $\frac{A}{x-a} + \frac{B}{x-b}$ |
| $(x-a)^2$ | $\frac{A}{x-a} + \frac{B}{(x-a)^2}$ |
| $(x-a)(x-b)(x-c)$ | $\frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}$ |
| $(x-a)^2(x-b)$ | $\frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}$ |
| $(x-a)(x^2+bx+c)$ | $\frac{A}{x-a} + \frac{Bx+C}{x^2+bx+c}$ |
Steps for Partial Fractions:
1. Check if degree of numerator < degree of denominator (if not, divide first)
2. Factorize the denominator completely
3. Write partial fractions according to factors
4. Find constants by equating coefficients or substituting values
5. Integrate each term separately
1. Check if degree of numerator < degree of denominator (if not, divide first)
2. Factorize the denominator completely
3. Write partial fractions according to factors
4. Find constants by equating coefficients or substituting values
5. Integrate each term separately
Integrals of Special Functions 🔥🔥
Type 1: $\sqrt{x^2 \pm a^2}$ and $\sqrt{a^2 - x^2}$
$\int \sqrt{x^2 - a^2} \, dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\log|x + \sqrt{x^2-a^2}| + C$
$\int \sqrt{x^2 + a^2} \, dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\log|x + \sqrt{x^2+a^2}| + C$
$\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$
Type 2: $\frac{px+q}{ax^2+bx+c}$
Method: Express $px + q = A\frac{d}{dx}(ax^2+bx+c) + B$
i.e., $px + q = A(2ax + b) + B$
Find A and B by comparing coefficients, then integrate
i.e., $px + q = A(2ax + b) + B$
Find A and B by comparing coefficients, then integrate
Type 3: $\frac{px+q}{\sqrt{ax^2+bx+c}}$
Method: Express $px + q = A\frac{d}{dx}(ax^2+bx+c) + B$
Then integrate each part separately using standard forms
Then integrate each part separately using standard forms
Quick Trick: For integrals involving $ax^2 + bx + c$:
1. Complete the square: $ax^2 + bx + c = a[(x + \frac{b}{2a})^2 \pm k^2]$
2. Substitute $t = x + \frac{b}{2a}$
3. Use standard formulas
1. Complete the square: $ax^2 + bx + c = a[(x + \frac{b}{2a})^2 \pm k^2]$
2. Substitute $t = x + \frac{b}{2a}$
3. Use standard formulas
Definite Integrals ★★★★★
Definition: If $F(x)$ is an antiderivative of $f(x)$, then:
$\int_a^b f(x)dx = F(b) - F(a) = [F(x)]_a^b$
This is called the Fundamental Theorem of Calculus
Properties of Definite Integrals
P0: $\int_a^b f(x)dx = \int_a^b f(t)dt$ (variable doesn't matter)
P1: $\int_a^b f(x)dx = -\int_b^a f(x)dx$
P2: $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$
P3: $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$ ⭐ Very Important!
P4: $\int_0^a f(x)dx = \int_0^a f(a-x)dx$ ⭐ Very Important!
P5: $\int_0^{2a} f(x)dx = \int_0^a f(x)dx + \int_0^a f(2a-x)dx$
P1: $\int_a^b f(x)dx = -\int_b^a f(x)dx$
P2: $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$
P3: $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$ ⭐ Very Important!
P4: $\int_0^a f(x)dx = \int_0^a f(a-x)dx$ ⭐ Very Important!
P5: $\int_0^{2a} f(x)dx = \int_0^a f(x)dx + \int_0^a f(2a-x)dx$
P6: $\int_0^{2a} f(x)dx = \begin{cases}
2\int_0^a f(x)dx & \text{if } f(2a-x) = f(x) \\
0 & \text{if } f(2a-x) = -f(x)
\end{cases}$
P7: $\int_{-a}^a f(x)dx = \begin{cases} 2\int_0^a f(x)dx & \text{if } f(-x) = f(x) \text{ (even)} \\ 0 & \text{if } f(-x) = -f(x) \text{ (odd)} \end{cases}$
P7: $\int_{-a}^a f(x)dx = \begin{cases} 2\int_0^a f(x)dx & \text{if } f(-x) = f(x) \text{ (even)} \\ 0 & \text{if } f(-x) = -f(x) \text{ (odd)} \end{cases}$
Strategy for Definite Integrals:
1. Check if function is even/odd → Use P7
2. Check if limits are 0 to 2a → Use P6
3. Try property P3 or P4 to simplify
4. If none work, evaluate normally using antiderivative
1. Check if function is even/odd → Use P7
2. Check if limits are 0 to 2a → Use P6
3. Try property P3 or P4 to simplify
4. If none work, evaluate normally using antiderivative
Solved Examples (Board Marking Scheme)
Q1. Evaluate: $\int \frac{2x}{(x^2+1)(x^2+3)} dx$ (3 Marks)
Step 1: Substitute 0.5 Mark
Let $x^2 = t$, then $2x \, dx = dt$
Integral becomes: $\int \frac{dt}{(t+1)(t+3)}$
Step 2: Partial Fractions 1 Mark
$\frac{1}{(t+1)(t+3)} = \frac{A}{t+1} + \frac{B}{t+3}$
$1 = A(t+3) + B(t+1)$
Put $t = -1$: $1 = 2A \Rightarrow A = \frac{1}{2}$
Put $t = -3$: $1 = -2B \Rightarrow B = -\frac{1}{2}$
Step 3: Integrate 1 Mark
$= \int \left(\frac{1/2}{t+1} - \frac{1/2}{t+3}\right) dt$
$= \frac{1}{2}\log|t+1| - \frac{1}{2}\log|t+3| + C$
Step 4: Back-substitute 0.5 Mark
$= \frac{1}{2}\log|x^2+1| - \frac{1}{2}\log|x^2+3| + C$
$= \frac{1}{2}\log\left|\frac{x^2+1}{x^2+3}\right| + C$
Q2. Evaluate: $\int x \sin x \, dx$ (2 Marks)
Solution: Integration by Parts 2 Marks
Using ILATE rule: Take $u = x$ (Algebraic) and $v = \sin x$ (Trigonometric)
$\int x \sin x \, dx = x \int \sin x \, dx - \int \left[\frac{dx}{dx} \cdot \int \sin x \, dx\right] dx$
$= x(-\cos x) - \int 1 \cdot (-\cos x) \, dx$
$= -x\cos x + \int \cos x \, dx$
$= -x\cos x + \sin x + C$
Q3. Evaluate: $\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$ (4 Marks)
Step 1: Apply Property 1 Mark
Let $I = \int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$ ... (1)
Using $\int_0^a f(x)dx = \int_0^a f(a-x)dx$:
$I = \int_0^{\pi/2} \frac{\sin(\pi/2 - x)}{\sin(\pi/2 - x) + \cos(\pi/2 - x)} dx$
Step 2: Simplify 1 Mark
$I = \int_0^{\pi/2} \frac{\cos x}{\cos x + \sin x} dx$ ... (2)
Step 3: Add Equations 1.5 Marks
Adding (1) and (2):
$2I = \int_0^{\pi/2} \frac{\sin x + \cos x}{\sin x + \cos x} dx$
$2I = \int_0^{\pi/2} 1 \, dx = [x]_0^{\pi/2} = \frac{\pi}{2}$
Step 4: Final Answer 0.5 Mark
$I = \frac{\pi}{4}$
Q4. Evaluate: $\int_0^1 \frac{x}{x^2 + 1} dx$ (2 Marks)
Solution: 2 Marks
Let $x^2 + 1 = t$, then $2x \, dx = dt$
When $x = 0$, $t = 1$; when $x = 1$, $t = 2$
$= \int_1^2 \frac{1}{2t} dt = \frac{1}{2}[\log t]_1^2$
$= \frac{1}{2}(\log 2 - \log 1) = \frac{1}{2}\log 2$
Previous Year Questions (PYQs)
2023 (1 Mark MCQ): $\int e^x(\tan x + 1)\sec x \, dx$ equals:
(A) $e^x \sec x + C$ (B) $e^x \tan x + C$ (C) $e^x \sin x + C$ (D) $e^x \cos x + C$
Ans: (A) $e^x \sec x + C$
Hint: Use $\int e^x[f(x) + f'(x)]dx = e^x f(x) + C$ where $f(x) = \sec x$
(A) $e^x \sec x + C$ (B) $e^x \tan x + C$ (C) $e^x \sin x + C$ (D) $e^x \cos x + C$
Ans: (A) $e^x \sec x + C$
Hint: Use $\int e^x[f(x) + f'(x)]dx = e^x f(x) + C$ where $f(x) = \sec x$
2023 (2 Marks): Evaluate: $\int \frac{x^2}{(x^2+1)(x^2+4)} dx$
Solution: Use partial fractions. Express as $\frac{x^2}{(x^2+1)(x^2+4)} = \frac{A}{x^2+1} + \frac{B}{x^2+4}$
After solving: $A = -\frac{4}{3}$, $B = \frac{1}{3}$
Answer: $-\frac{4}{3}\tan^{-1}x + \frac{1}{6}\tan^{-1}\frac{x}{2} + C$
Solution: Use partial fractions. Express as $\frac{x^2}{(x^2+1)(x^2+4)} = \frac{A}{x^2+1} + \frac{B}{x^2+4}$
After solving: $A = -\frac{4}{3}$, $B = \frac{1}{3}$
Answer: $-\frac{4}{3}\tan^{-1}x + \frac{1}{6}\tan^{-1}\frac{x}{2} + C$
2022 (3 Marks): Evaluate: $\int_0^{\pi/4} \log(1 + \tan x) dx$
Hint: Use property $\int_0^a f(x)dx = \int_0^a f(a-x)dx$
After applying: $I = \int_0^{\pi/4} \log(1 + \tan(\frac{\pi}{4}-x)) dx = \int_0^{\pi/4} \log 2 dx$
Answer: $\frac{\pi}{4}\log 2$
Hint: Use property $\int_0^a f(x)dx = \int_0^a f(a-x)dx$
After applying: $I = \int_0^{\pi/4} \log(1 + \tan(\frac{\pi}{4}-x)) dx = \int_0^{\pi/4} \log 2 dx$
Answer: $\frac{\pi}{4}\log 2$
2022 (4 Marks): Evaluate: $\int \frac{x^2 + 1}{x^4 + 1} dx$
Hint: Divide numerator and denominator by $x^2$, then substitute $x - \frac{1}{x} = t$
Answer: $\frac{1}{\sqrt{2}}\tan^{-1}\frac{x^2-1}{\sqrt{2}x} + C$
Hint: Divide numerator and denominator by $x^2$, then substitute $x - \frac{1}{x} = t$
Answer: $\frac{1}{\sqrt{2}}\tan^{-1}\frac{x^2-1}{\sqrt{2}x} + C$
2021 (4 Marks): Evaluate: $\int_0^{\pi} x\sin x \cos^2 x \, dx$
Solution: Use $\int_0^a x f(x)dx = a\int_0^a f(x)dx$ (if applicable) or integration by parts
Answer: $\frac{\pi}{4}$
Solution: Use $\int_0^a x f(x)dx = a\int_0^a f(x)dx$ (if applicable) or integration by parts
Answer: $\frac{\pi}{4}$
2020 (6 Marks): Evaluate: $\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx$
Solution: $\sin^7 x$ is an odd function, i.e., $f(-x) = -f(x)$
Using property P7: $\int_{-a}^a f(x)dx = 0$ if $f$ is odd
Answer: $0$
Solution: $\sin^7 x$ is an odd function, i.e., $f(-x) = -f(x)$
Using property P7: $\int_{-a}^a f(x)dx = 0$ if $f$ is odd
Answer: $0$
Exam Strategy & Mistake Bank
Common Mistakes 🚨
Mistake 1: Forgetting to add constant of integration (+C) in indefinite integrals. This costs 0.5-1 mark!
Mistake 2: Not changing limits when doing substitution in definite integrals. Either change limits or back-substitute!
Mistake 3: Using wrong ILATE order in integration by parts. Always: I → L → A → T → E
Mistake 4: Forgetting absolute value in $\log|x|$. Without |x|, you may lose marks!
Mistake 5: Not simplifying before integration. Always simplify algebraic expressions first!
Mistake 6: Mixing up $\int \frac{1}{x^2+a^2}$ and $\int \frac{1}{x^2-a^2}$ formulas. One gives $\tan^{-1}$, other gives $\log$!
Scoring Tips 🏆
Tip 1: For definite integrals, ALWAYS check if the function is even/odd first. Can save 3-4 minutes!
Tip 2: Show all substitution steps clearly: "Let t = ..., then dt = ..." for full marks.
Tip 3: In integration by parts, clearly mark which is u and which is v.
Tip 4: For partial fractions, show the step where you find A, B, C values by substitution or comparison.
Tip 5: If asked to evaluate $\int_a^b f(x)dx$, write the antiderivative first, then substitute limits with proper notation: $[F(x)]_a^b$
Tip 6: For trigonometric integrals, use identities to simplify before integrating: $\sin^2 x = \frac{1-\cos 2x}{2}$
Important Theorems & Results
First Fundamental Theorem of Integral Calculus:
Let $A(x) = \int_a^x f(t)dt$. Then $A'(x) = f(x)$ for all $x \in [a,b]$
Let $A(x) = \int_a^x f(t)dt$. Then $A'(x) = f(x)$ for all $x \in [a,b]$
Second Fundamental Theorem of Integral Calculus:
Let $f$ be continuous on $[a,b]$ and $F$ be an antiderivative of $f$. Then: $\int_a^b f(x)dx = F(b) - F(a)$
Let $f$ be continuous on $[a,b]$ and $F$ be an antiderivative of $f$. Then: $\int_a^b f(x)dx = F(b) - F(a)$
Important Results:
1. $\frac{d}{dx}\int_a^x f(t)dt = f(x)$
2. $\frac{d}{dx}\int_a^{g(x)} f(t)dt = f(g(x)) \cdot g'(x)$
3. If $f$ is even and $g$ is odd, then $\int_{-a}^a f(x)g(x)dx = 0$
4. $\int_0^{\pi/2} f(\sin x)dx = \int_0^{\pi/2} f(\cos x)dx$
5. $\int_0^{\pi} x f(\sin x)dx = \frac{\pi}{2}\int_0^{\pi} f(\sin x)dx$
1. $\frac{d}{dx}\int_a^x f(t)dt = f(x)$
2. $\frac{d}{dx}\int_a^{g(x)} f(t)dt = f(g(x)) \cdot g'(x)$
3. If $f$ is even and $g$ is odd, then $\int_{-a}^a f(x)g(x)dx = 0$
4. $\int_0^{\pi/2} f(\sin x)dx = \int_0^{\pi/2} f(\cos x)dx$
5. $\int_0^{\pi} x f(\sin x)dx = \frac{\pi}{2}\int_0^{\pi} f(\sin x)dx$
Walli's Formula (Advanced)
$\int_0^{\pi/2} \sin^n x \, dx = \int_0^{\pi/2} \cos^n x \, dx = \begin{cases}
\frac{(n-1)(n-3)...2}{n(n-2)...3} \cdot \frac{\pi}{2} & \text{if } n \text{ is odd} \\
\frac{(n-1)(n-3)...1}{n(n-2)...2} & \text{if } n \text{ is even}
\end{cases}$
Practice Problems (Self-Assessment)
Level 1: Basic (1-2 Marks Each)
Q1. Evaluate: $\int (3x^2 + 2x + 1) dx$
Answer: $x^3 + x^2 + x + C$
Q2. Evaluate: $\int \frac{1}{x^2 + 9} dx$
Answer: $\frac{1}{3}\tan^{-1}\frac{x}{3} + C$
Q3. Evaluate: $\int_0^1 x e^{x^2} dx$
Hint: Substitute $x^2 = t$. Answer: $\frac{e-1}{2}$
Level 2: Intermediate (3-4 Marks Each)
Q4. Evaluate: $\int \frac{x+2}{x^2+5x+6} dx$
Hint: Use partial fractions after factorizing denominator
Q5. Evaluate: $\int x^2 \sin x \, dx$
Hint: Use integration by parts twice
Q6. Evaluate: $\int_0^{\pi/4} \tan^2 x \, dx$
Hint: $\tan^2 x = \sec^2 x - 1$
Level 3: Advanced (5-6 Marks Each)
Q7. Evaluate: $\int_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx$
Hint: Use $\int_0^a x f(x)dx = \int_0^a (a-x)f(x)dx$
Q8. Evaluate: $\int \frac{x^3}{\sqrt{1+x^2}} dx$
Hint: Write $x^3 = x^2 \cdot x$ and substitute $1+x^2 = t^2$
Q9. Evaluate: $\int_{-\pi/2}^{\pi/2} \frac{x^3 \cos^2 x}{1+e^x} dx$
Hint: Check if function is odd/even and use appropriate property
Formula Sheet (Must Remember!) 📝
Standard Integrals
Basic Forms:
1. $\int x^n dx = \frac{x^{n+1}}{n+1} + C$
2. $\int \frac{1}{x} dx = \log|x| + C$
3. $\int e^x dx = e^x + C$
4. $\int a^x dx = \frac{a^x}{\log a} + C$
5. $\int \sin x dx = -\cos x + C$
6. $\int \cos x dx = \sin x + C$
7. $\int \tan x dx = \log|\sec x| + C$
8. $\int \cot x dx = \log|\sin x| + C$
1. $\int x^n dx = \frac{x^{n+1}}{n+1} + C$
2. $\int \frac{1}{x} dx = \log|x| + C$
3. $\int e^x dx = e^x + C$
4. $\int a^x dx = \frac{a^x}{\log a} + C$
5. $\int \sin x dx = -\cos x + C$
6. $\int \cos x dx = \sin x + C$
7. $\int \tan x dx = \log|\sec x| + C$
8. $\int \cot x dx = \log|\sin x| + C$
Advanced Forms:
9. $\int \sec x dx = \log|\sec x + \tan x| + C$
10. $\int \text{cosec} x dx = \log|\text{cosec} x - \cot x| + C$
11. $\int \sec^2 x dx = \tan x + C$
12. $\int \text{cosec}^2 x dx = -\cot x + C$
13. $\int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C$
14. $\int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\frac{x}{a} + C$
15. $\int \frac{dx}{x^2-a^2} = \frac{1}{2a}\log|\frac{x-a}{x+a}| + C$
9. $\int \sec x dx = \log|\sec x + \tan x| + C$
10. $\int \text{cosec} x dx = \log|\text{cosec} x - \cot x| + C$
11. $\int \sec^2 x dx = \tan x + C$
12. $\int \text{cosec}^2 x dx = -\cot x + C$
13. $\int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C$
14. $\int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\frac{x}{a} + C$
15. $\int \frac{dx}{x^2-a^2} = \frac{1}{2a}\log|\frac{x-a}{x+a}| + C$
Special Integrals
16. $\int \sqrt{x^2-a^2} dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\log|x+\sqrt{x^2-a^2}| + C$17. $\int \sqrt{x^2+a^2} dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\log|x+\sqrt{x^2+a^2}| + C$
18. $\int \sqrt{a^2-x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$
19. $\int e^x[f(x) + f'(x)]dx = e^x f(x) + C$
Properties of Definite Integrals
P1: $\int_a^b f(x)dx = -\int_b^a f(x)dx$P2: $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$
P3: $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$ ⭐
P4: $\int_0^a f(x)dx = \int_0^a f(a-x)dx$ ⭐
P7: $\int_{-a}^a f(x)dx = 2\int_0^a f(x)dx$ if f is even, $= 0$ if f is odd ⭐⭐⭐
Quick Revision Notes (Night Before Exam)
Integration Techniques - When to Use What?
| If You See... | Method to Use | Example |
|---|---|---|
| Function and its derivative together | Substitution | $\int 2x e^{x^2} dx$ (let $x^2 = t$) |
| Product of two different types | Integration by Parts (ILATE) | $\int x \sin x dx$ |
| Rational function (proper) | Partial Fractions | $\int \frac{1}{(x+1)(x+2)} dx$ |
| $e^x[f(x) + f'(x)]$ | Direct Formula | $\int e^x(\tan x + \sec^2 x) dx = e^x \tan x + C$ |
| $\sqrt{a^2-x^2}$, $\sqrt{x^2+a^2}$, $\sqrt{x^2-a^2}$ | Trigonometric Substitution | $x = a\sin\theta$, $x = a\tan\theta$, $x = a\sec\theta$ |
| $\frac{px+q}{ax^2+bx+c}$ | Express as $A\frac{d}{dx}(denominator) + B$ | $\int \frac{2x+3}{x^2+x+1} dx$ |
Common Definite Integral Tricks
Trick 1: For $\int_0^a f(x)dx$, try replacing x with (a-x) and add both
Trick 2: For $\int_{-a}^a f(x)dx$, check if f is even or odd
Trick 3: For $\int_0^{2a} f(x)dx$, split into $\int_0^a + \int_a^{2a}$ and use substitution
Trick 4: For $\int_0^{\pi/2}$ with sin and cos, they're interchangeable: $\int_0^{\pi/2} \sin x dx = \int_0^{\pi/2} \cos x dx$
Trick 5: $\int_0^{\pi} x f(\sin x) dx = \frac{\pi}{2} \int_0^{\pi} f(\sin x) dx$ (very useful!)
Trick 2: For $\int_{-a}^a f(x)dx$, check if f is even or odd
Trick 3: For $\int_0^{2a} f(x)dx$, split into $\int_0^a + \int_a^{2a}$ and use substitution
Trick 4: For $\int_0^{\pi/2}$ with sin and cos, they're interchangeable: $\int_0^{\pi/2} \sin x dx = \int_0^{\pi/2} \cos x dx$
Trick 5: $\int_0^{\pi} x f(\sin x) dx = \frac{\pi}{2} \int_0^{\pi} f(\sin x) dx$ (very useful!)
Time-Saving Calculator Checks
1. For indefinite integrals, differentiate your answer to check
2. For definite integrals with limits 0 to $\pi/2$ or $-a$ to $a$, first check properties
3. If integral looks very complicated, there's usually a property that simplifies it
4. Remember: $\int_a^a f(x)dx = 0$ always!
5. For symmetry: Draw a quick graph to see if function is even/odd
2. For definite integrals with limits 0 to $\pi/2$ or $-a$ to $a$, first check properties
3. If integral looks very complicated, there's usually a property that simplifies it
4. Remember: $\int_a^a f(x)dx = 0$ always!
5. For symmetry: Draw a quick graph to see if function is even/odd
Exam Day Strategy (90 Minutes Plan)
What to Attempt First
- All MCQs (1 mark each) - 5 mins
- Direct formula questions (2 marks) - 10 mins
- Standard substitution problems (3 marks) - 15 mins
- Definite integral with properties (4 marks) - 20 mins
- Integration by parts (4-6 marks) - 25 mins
- Complex problems (6 marks) - 15 mins
Marks Distribution Strategy
Easy (70% marks):
- Standard formulas
- Simple substitution
- Integration by parts (one iteration)
- Definite integrals with odd/even
Medium (20% marks):
- Partial fractions
- Complex substitutions
- Multiple property applications
Difficult (10% marks):
- New pattern questions
- Multi-step problems
⚠️ Critical Reminders:
• Write +C in every indefinite integral (0.5 mark deduction if missing)
• Show substitution clearly: "Let t = ..., then dt = ..."
• For definite integrals, write limits with bracket notation: $[F(x)]_a^b$
• Check dimensional consistency in physics-related integrals
• Use absolute value in logarithms: $\log|x|$ not $\log x$
• Write +C in every indefinite integral (0.5 mark deduction if missing)
• Show substitution clearly: "Let t = ..., then dt = ..."
• For definite integrals, write limits with bracket notation: $[F(x)]_a^b$
• Check dimensional consistency in physics-related integrals
• Use absolute value in logarithms: $\log|x|$ not $\log x$
Special Cases & Exceptions (High Weightage)
Case 1: Rational Functions with Quadratic Denominators
Type: $\int \frac{px+q}{ax^2+bx+c} dx$
Method: Express $px + q = A(2ax + b) + B$ where $2ax + b$ is derivative of denominator
Example: $\int \frac{3x+1}{x^2+2x+5} dx$
Write $3x + 1 = A(2x + 2) + B \Rightarrow A = \frac{3}{2}, B = -2$
Method: Express $px + q = A(2ax + b) + B$ where $2ax + b$ is derivative of denominator
Example: $\int \frac{3x+1}{x^2+2x+5} dx$
Write $3x + 1 = A(2x + 2) + B \Rightarrow A = \frac{3}{2}, B = -2$
Case 2: Integrals of Type $\int \frac{dx}{(x-a)(x-b)}$
$\int \frac{dx}{(x-a)(x-b)} = \frac{1}{b-a}\log\left|\frac{x-b}{x-a}\right| + C$
Case 3: Reduction Formulas
For $\int \sin^n x dx$ or $\int \cos^n x dx$:
Use: $I_n = -\frac{1}{n}\sin^{n-1}x \cos x + \frac{n-1}{n}I_{n-2}$
But usually easier to use trigonometric identities!
Use: $I_n = -\frac{1}{n}\sin^{n-1}x \cos x + \frac{n-1}{n}I_{n-2}$
But usually easier to use trigonometric identities!
Case 4: Logarithmic Integrals
$\int \log x \, dx = x \log x - x + C$
$\int (\log x)^2 dx = x[(\log x)^2 - 2\log x + 2] + C$
Memory Trick for $\int \log x dx$:
Think: "x log x minus x" (sounds like a phrase!)
Think: "x log x minus x" (sounds like a phrase!)
MCQ Strategies & Shortcuts
Type 1: Matching Derivatives
Quick Method: Differentiate the options and match with the integrand!
Example: $\int 2x e^{x^2} dx = ?$
Check: $\frac{d}{dx}(e^{x^2}) = 2x e^{x^2}$ ✓
Answer: $e^{x^2} + C$
Example: $\int 2x e^{x^2} dx = ?$
Check: $\frac{d}{dx}(e^{x^2}) = 2x e^{x^2}$ ✓
Answer: $e^{x^2} + C$
Type 2: Definite Integral Properties
Quick Checks:
1. Is the function odd? If yes and limits are $-a$ to $a$, answer is 0!
2. Can you use $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$?
3. For $\int_0^{\pi/2}$, sin and cos are interchangeable
1. Is the function odd? If yes and limits are $-a$ to $a$, answer is 0!
2. Can you use $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$?
3. For $\int_0^{\pi/2}$, sin and cos are interchangeable
Type 3: Elimination Method
Eliminate by:
• Checking dimensions/units
• Substituting x = 0 or x = 1 in both integrand and options
• Checking signs (is result positive/negative?)
• Differentiation test (fastest!)
• Checking dimensions/units
• Substituting x = 0 or x = 1 in both integrand and options
• Checking signs (is result positive/negative?)
• Differentiation test (fastest!)
Practice MCQ: $\int e^x(\sin x + \cos x) dx = ?$
(A) $e^x \sin x + C$ (B) $e^x \cos x + C$ (C) $e^x(\sin x - \cos x) + C$ (D) $e^x(\sin x + \cos x) + C$
Solution: Use $\int e^x[f(x) + f'(x)]dx = e^x f(x) + C$
Here $f(x) = \sin x$, $f'(x) = \cos x$
Answer: (A) $e^x \sin x + C$
(A) $e^x \sin x + C$ (B) $e^x \cos x + C$ (C) $e^x(\sin x - \cos x) + C$ (D) $e^x(\sin x + \cos x) + C$
Solution: Use $\int e^x[f(x) + f'(x)]dx = e^x f(x) + C$
Here $f(x) = \sin x$, $f'(x) = \cos x$
Answer: (A) $e^x \sin x + C$
Final Checklist (Print & Keep)
✓ Before Entering Exam Hall
Formulas to Recite:
- All standard integrals (15 formulas)
- Integration by parts formula
- 3 special integral formulas ($\sqrt{x^2 \pm a^2}$)
- 5 definite integral properties
- $\int e^x[f(x) + f'(x)]dx$ formula
Concepts to Recall:
- ILATE rule order
- When to use which substitution
- Partial fraction types
- Even vs Odd function test
- Common mistakes (absolute value, +C)
✓ During Exam
- ✓ Read question twice, identify the type
- ✓ For definite integrals, check properties FIRST
- ✓ Write all steps clearly (show substitution, limits)
- ✓ Always add +C for indefinite integrals
- ✓ Use proper notation: $[F(x)]_a^b = F(b) - F(a)$
- ✓ Verify answer by differentiation if time permits
- ✓ Check sign of final answer (should it be positive?)
Golden Rules for 20/20 in Integrals
1. Master all 15 standard formulas - No excuses!2. Practice 10 definite integrals using properties daily
3. Solve at least 5 PYQs from each type
4. Time yourself: 2 marks = 4 mins, 4 marks = 8 mins
5. Never leave a question - attempt with what you know!