Exercise 7.1 Class 12 Maths Integrals

NCERT Solutions Class 12 Maths Ex 7.1 | Full Exercise

💡 Integration Formulas

Key formulas used in this exercise:

[Image of basic integration formulas table]

$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$
$\int e^{ax} dx = \frac{e^{ax}}{a} + C$
$\int \sin(ax) dx = -\frac{\cos(ax)}{a} + C$
$\int \cos(ax) dx = \frac{\sin(ax)}{a} + C$

Questions 01 — 05

Find an anti-derivative using the Method of Inspection.

1. $\sin 2x$

$$\frac{d}{dx}(\cos 2x) = -2\sin 2x \implies \frac{d}{dx}\left(-\frac{1}{2}\cos 2x\right) = \sin 2x$$

Ans: $-\frac{1}{2}\cos 2x$

2. $\cos 3x$

$$\frac{d}{dx}(\sin 3x) = 3\cos 3x \implies \frac{d}{dx}\left(\frac{1}{3}\sin 3x\right) = \cos 3x$$

Ans: $\frac{1}{3}\sin 3x$

3. $e^{2x}$

$$\frac{d}{dx}(e^{2x}) = 2e^{2x} \implies \frac{d}{dx}\left(\frac{1}{2}e^{2x}\right) = e^{2x}$$

Ans: $\frac{1}{2}e^{2x}$

4. $(ax + b)^2$

$$\frac{d}{dx}(ax+b)^3 = 3a(ax+b)^2 \implies \frac{d}{dx}\left(\frac{1}{3a}(ax+b)^3\right) = (ax+b)^2$$

Ans: $\frac{1}{3a}(ax+b)^3$

5. $\sin 2x – 4e^{3x}$

Combines Q1 and Q3 logic: Anti-derivative is $-\frac{1}{2}\cos 2x – 4(\frac{1}{3}e^{3x})$

Ans: $-\frac{1}{2}\cos 2x – \frac{4}{3}e^{3x}$

Question 06

Find $\int (4e^{3x} + 1) dx$

$$I = 4\int e^{3x} dx + \int 1 dx = 4\left(\frac{e^{3x}}{3}\right) + x + C$$

Result: $\frac{4}{3}e^{3x} + x + C$

Question 07

Find $\int x^2 (1 – \frac{1}{x^2}) dx$

Step 1: Simplify

$$x^2(1 – \frac{1}{x^2}) = x^2 – x^2(\frac{1}{x^2}) = x^2 – 1$$

Step 2: Integrate

$$\int (x^2 – 1) dx = \frac{x^3}{3} – x + C$$

Result: $\frac{x^3}{3} – x + C$

Question 08

Find $\int (ax^2 + bx + c) dx$

$$I = a\int x^2 dx + b\int x dx + c\int 1 dx = \frac{ax^3}{3} + \frac{bx^2}{2} + cx + C$$

Result: $\frac{ax^3}{3} + \frac{bx^2}{2} + cx + C$

Question 09

Find $\int (2x^2 + e^x) dx$

$$I = 2\int x^2 dx + \int e^x dx = \frac{2}{3}x^3 + e^x + C$$

Result: $\frac{2}{3}x^3 + e^x + C$

Question 10

Find $\int (\sqrt{x} – \frac{1}{\sqrt{x}})^2 dx$

Step 1: Expand $(a-b)^2$

$$(\sqrt{x})^2 + \left(\frac{1}{\sqrt{x}}\right)^2 – 2(\sqrt{x})\left(\frac{1}{\sqrt{x}}\right) = x + \frac{1}{x} – 2$$

Step 2: Integrate

$$I = \int x dx + \int \frac{1}{x} dx – \int 2 dx = \frac{x^2}{2} + \log|x| – 2x + C$$

Result: $\frac{x^2}{2} + \log|x| – 2x + C$

Question 11

Find $\int \frac{x^3 + 5x^2 – 4}{x^2} dx$

Step 1: Divide

$$\frac{x^3}{x^2} + \frac{5x^2}{x^2} – \frac{4}{x^2} = x + 5 – 4x^{-2}$$

Step 2: Integrate

$$I = \frac{x^2}{2} + 5x – 4\left(\frac{x^{-1}}{-1}\right) + C = \frac{x^2}{2} + 5x + \frac{4}{x} + C$$

Result: $\frac{x^2}{2} + 5x + \frac{4}{x} + C$

Question 12

Find $\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx$

Step 1: Simplify

$$\frac{x^3}{x^{1/2}} + \frac{3x}{x^{1/2}} + \frac{4}{x^{1/2}} = x^{5/2} + 3x^{1/2} + 4x^{-1/2}$$

Step 2: Integrate

$$I = \frac{x^{7/2}}{7/2} + 3\frac{x^{3/2}}{3/2} + 4\frac{x^{1/2}}{1/2} + C$$ $$I = \frac{2}{7}x^{7/2} + 2x^{3/2} + 8\sqrt{x} + C$$

Question 13

Find $\int \frac{x^3 – x^2 + x – 1}{x – 1} dx$

Step 1: Factorize

$$x^2(x-1) + 1(x-1) = (x^2+1)(x-1)$$ $$\text{Integrand becomes } \frac{(x^2+1)(x-1)}{x-1} = x^2 + 1$$

Step 2: Integrate

$$\int (x^2 + 1) dx = \frac{x^3}{3} + x + C$$

Question 14

Find $\int (1-x)\sqrt{x} dx$

Step 1: Expand

$$\sqrt{x} – x\sqrt{x} = x^{1/2} – x^{3/2}$$

Step 2: Integrate

$$I = \frac{x^{3/2}}{3/2} – \frac{x^{5/2}}{5/2} + C = \frac{2}{3}x^{3/2} – \frac{2}{5}x^{5/2} + C$$

Question 15

Find $\int \sqrt{x}(3x^2 + 2x + 3) dx$

Step 1: Multiply

$$3x^{5/2} + 2x^{3/2} + 3x^{1/2}$$

Step 2: Integrate

$$I = 3\left(\frac{2}{7}x^{7/2}\right) + 2\left(\frac{2}{5}x^{5/2}\right) + 3\left(\frac{2}{3}x^{3/2}\right) + C$$ $$I = \frac{6}{7}x^{7/2} + \frac{4}{5}x^{5/2} + 2x^{3/2} + C$$

Question 16

Find $\int (2x – 3\cos x + e^x) dx$

$$I = 2\frac{x^2}{2} – 3\sin x + e^x + C = x^2 – 3\sin x + e^x + C$$

Question 17

Find $\int (2x^2 – 3\sin x + 5\sqrt{x}) dx$

$$I = 2\frac{x^3}{3} – 3(-\cos x) + 5\frac{2}{3}x^{3/2} + C$$ $$I = \frac{2}{3}x^3 + 3\cos x + \frac{10}{3}x^{3/2} + C$$

Question 18

Find $\int \sec x (\sec x + \tan x) dx$

Step 1: Expand

$$\sec^2 x + \sec x \tan x$$

Step 2: Integrate

$$\int \sec^2 x dx + \int \sec x \tan x dx = \tan x + \sec x + C$$

Question 19

Find $\int \frac{\sec^2 x}{\text{cosec}^2 x} dx$

Step 1: Simplify Trigonometry

$$\frac{1/\cos^2 x}{1/\sin^2 x} = \frac{\sin^2 x}{\cos^2 x} = \tan^2 x$$

Step 2: Use Identity

$$\int (\sec^2 x – 1) dx = \tan x – x + C$$

Question 20

Find $\int \frac{2 – 3\sin x}{\cos^2 x} dx$

Step 1: Split Numerator

$$\frac{2}{\cos^2 x} – \frac{3\sin x}{\cos^2 x} = 2\sec^2 x – 3\tan x \sec x$$

Step 2: Integrate

$$2\tan x – 3\sec x + C$$

Question 21 • MCQ

The anti-derivative of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ equals…

$$I = \int (x^{1/2} + x^{-1/2}) dx = \frac{x^{3/2}}{3/2} + \frac{x^{1/2}}{1/2} + C = \frac{2}{3}x^{3/2} + 2x^{1/2} + C$$

Correct Option: (C)

Question 22 • MCQ

If $\frac{d}{dx}f(x) = 4x^3 – \frac{3}{x^4}$ such that $f(2) = 0$, find $f(x)$.

Step 1: Find f(x) by integration

$$f(x) = \int (4x^3 – 3x^{-4}) dx = x^4 – 3\frac{x^{-3}}{-3} + C = x^4 + \frac{1}{x^3} + C$$

Step 2: Find C using f(2) = 0

$$f(2) = 2^4 + \frac{1}{2^3} + C = 0 \implies 16 + \frac{1}{8} + C = 0 \implies C = -\frac{129}{8}$$

Correct Option: (A) $x^4 + \frac{1}{x^3} – \frac{129}{8}$