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.

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

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.
e^2x
Answer:
Let I = ∫e^2x dx
We know that

Question 4.
(ax + b)²
Answer:
Let I = ∫(ax + b)² dx
We know that

Question 5.
sin 2x - 4e^3x
Answer:
Let I = ∫ (sin 2x - 4e^3x) dx
= ∫ sin 2x dx - 4 ∫e^3x dx

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

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

Question 10.
∫ \(\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2}\) dx
Answer:

Question 11.
∫ \(\frac{x^{3}+5 x^{2}-4}{x^{2}}\) dx
Answer:

Question 12.
∫\(\frac{x^{3}+3 x+4}{\sqrt{x}}\) dx
Answer:

Question 13.
∫\(\frac{x^{3}-x^{2}+x-1}{x-1}\) dx
Answer:

Question 14.
∫ (1 - x) √x dx
Answer:
∫ (1 - x) √x dx = ∫(√x - x√x)dx
= ∫ (x^1/2 - x.x^1/2) dx
= ∫ x^1/2 dx - ∫ x^3/2 dx

Question 15.
∫ √x(3x² + 2x + 3) dx
Answer:

Question 16.
∫ (2x - 3 cos x + e^x) dx
Answer:
∫ (2x - 3 cos x + e^x) dx
= 2 ∫x dx - 3 ∫ cos x dx + ∫ e^x dx
= 2.\(\frac{x^{2}}{2}\) - 3 sin x + e^x + C
= x² - 3 sin x + e^x + 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 ∫x^1/2 dx

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

Question 21.
The antiderivative of (√x + \(\frac{1}{\sqrt{x}}\)) equals:
(A) \(\frac{1}{3}\)x^1/3 + 2x^1/2 + C
(B) \(\frac{2}{3}\)x^2/3 + \(\frac{1}{2}\)x² + C
(C) \(\frac{2}{3}\)x^3/2 + 2x^1/2 + C
(D) \(\frac{3}{2}\)x^3/2 + \(\frac{1}{2}\)x^1/2 + C
Answer:

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: