RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 Textbook Exercise Questions and Answers.

Rajasthan Board RBSE Solutions for Class 12 Maths in Hindi Medium & English Medium are part of RBSE Solutions for Class 12. Students can also read RBSE Class 12 Maths Important Questions for exam preparation. Students can also go through RBSE Class 12 Maths Notes to understand and remember the concepts easily.

RBSE Class 12 Maths Solutions Chapter 7 Integrals Ex 7.1

Question 1.
sin 2x
Answer:
Let I = ∫ sin 2x dx
We know that
\(\frac{d}{d x}\) (cos 2x) = - 2 sin 2x
\(\frac{d}{d x}\) (-\(\frac{1}{2}\)cos 2x) = sin 2x
∴ ∫sin 2x dx = - \(\frac{1}{2}\) cos 2x + C

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

Question 2.
cos 3x
Answer:
Let I = cos 3x
We know that
\(\frac{d}{d x}\) (sin 3x) = 3 cos 3x
\(\frac{d}{d x}\) (-\(\frac{1}{3}\)sin 3x) = cos 3x
∴ ∫cos 3x dx = - \(\frac{1}{3}\) sin 3x + C

Question 3.
e2x
Answer:
Let I = ∫e2x dx
We know that
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 1

Question 4.
(ax + b)2
Answer:
Let I = ∫(ax + b)2 dx
We know that
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 2

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

Question 5.
sin 2x - 4e3x
Answer:
Let I = ∫ (sin 2x - 4e3x) dx
= ∫ sin 2x dx - 4 ∫e3x dx
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 3

Question 6.
∫ (4e3x + 1) dx
Answer:
∫ (4e3x + 1) dx = 4 ∫e3x dx + ∫ 1 dx
= 4 \(\frac{e^{3 x}}{3}\) + x + C
= \(\frac{4}{3}\) e3x + x + C

Question 7.
∫x2\(\left(1-\frac{1}{x^{2}}\right)\) dx
Answer:
Let I = ∫x2\(\left(1-\frac{1}{x^{2}}\right) \)dx
= ∫ (x2 - 1) dx
= ∫ x2 dx - ∫ 1 dx
= \(\frac{x^{3}}{3}\) - x + C

Question 8.
∫ (ax2 + bx + c) dx
Answer:
∫ (ax2 + bx + c) dx
= a ∫x2 dx + b ∫ x dx + c ∫ 1.dx
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 4

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

Question 9.
∫ (2x2 + ex) dx
Answer:
∫ (2x2 + ex) dx = 2 ∫ x2 dx + ∫ ex dx
= 2\(\frac{x^{2+1}}{2+1}\) + ex + C
= \(\frac{2}{3}\)x3 + ex + C

Question 10.
\(\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2}\) dx
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 5

Question 11.
\(\frac{x^{3}+5 x^{2}-4}{x^{2}}\) dx
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 6

Question 12.
\(\frac{x^{3}+3 x+4}{\sqrt{x}}\) dx
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 7

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

Question 13.
\(\frac{x^{3}-x^{2}+x-1}{x-1}\) dx
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 8

Question 14.
∫ (1 - x) √x dx
Answer:
∫ (1 - x) √x dx = ∫(√x - x√x)dx
= ∫ (x1/2 - x.x1/2) dx
= ∫ x1/2 dx - ∫ x3/2 dx
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 9

Question 15.
∫ √x(3x2 + 2x + 3) dx
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 10

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

Question 16.
∫ (2x - 3 cos x + ex) dx
Answer:
∫ (2x - 3 cos x + ex) dx
= 2 ∫x dx - 3 ∫ cos x dx + ∫ ex dx
= 2.\(\frac{x^{2}}{2}\) - 3 sin x + ex + C
= x2 - 3 sin x + ex + C

Question 17.
∫ (2x2 - 3 sin x + 5√x) dx
Answer:
∫ (2x2 - 3 sin x + 5√x) dx
= 2 ∫x2 dx - 3 ∫ sin x dx + 5 ∫x1/2 dx
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 11

Question 18.
∫ sec x (sec x + tan x) dx
Answer:
∫ sec x (sec x + tan x) dx
= ∫ sec2x dx + ∫sec x tan x dx
= tan x + sec x + C

Question 19.
\(\frac{\sec ^{2} x}{{cosec}^{2} x}\)
Answer:
Let I = ∫\(\frac{\sec ^{2} x}{{cosec}^{2} x}\) dx = ∫\(\frac{\sin ^{2} x}{\cos ^{2} x}\) dx
= ∫tan2 dx
= ∫ (sec2 x - 1) dx
= ∫ sec2 x dx - ∫1. dx
= tan x - x + C

Question 20.
\(\frac{2-3 \sin x}{\cos ^{2} x}\) dx
Answer:
Let I = ∫\(\frac{2-3 \sin x}{\cos ^{2} x}\) dx
= ∫\(\frac{2}{\cos ^{2} x} dx - 3∫\frac{\sin x}{\cos ^{2} x}\) dx
= 2 ∫ sec2 x dx - 3 ∫ sec x tan x dx
= 2 tan x - 3 sec x + C

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

Question 21.
The antiderivative of (√x + \(\frac{1}{\sqrt{x}}\)) equals:
(A) \(\frac{1}{3}\)x1/3 + 2x1/2 + C
(B) \(\frac{2}{3}\)x2/3 + \(\frac{1}{2}\)x2 + C
(C) \(\frac{2}{3}\)x3/2 + 2x1/2 + C
(D) \(\frac{3}{2}\)x3/2 + \(\frac{1}{2}\)x1/2 + C
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 12
Hence, (C) is the correct answer.

Question 22.
If \(\frac{d}{d x}\) f(x) = 4x3 - \(\frac{3}{x^{4}}\) such that f(2) = 0. Then f(x) is:
(A) \(x^{4}+\frac{1}{x^{3}}-\frac{129}{8}\)
(B) \(x^{3}+\frac{1}{x^{4}}+\frac{129}{8}\)
(C) \(x^{4}+\frac{1}{x^{3}}+\frac{129}{8}\)
(D) \(x^{3}+\frac{1}{x^{4}}-\frac{129}{8}\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 13

Bhagya
Last Updated on Nov. 3, 2023, 9:21 a.m.
Published Nov. 2, 2023