RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 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 7.6

Question 1.
x sin x
Answer:
​​ RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 1
= x( - cos x) - ∫[\(\frac{d}{d x}\) x ∫sin x dx] dx
= x (- cos x) - ∫ [1.(- cos x) dx]
= - x cos x + ∫ cos x dx
= - x cos x + sin x + C

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 2.
x sin 3x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 2

Question 3.
x2 ex
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 3
= x2 ex - 2[xex - ∫1.ex dx]
= x2ex - 2[xex - ex] + C
= x2ex - 2xex - 2ex + C
= ex (x2 - 2x + 2) + C

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 4.
x log x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 4

Question 5.
x log 2x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 5

Question 6.
x2 log x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 6

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 7.
x sin-1 x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 7
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 8

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 8.
x tan-1 x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 9

Question 9.
x cos-1 x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 10

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 10.
(sin-1 x)2
Answer:
Let I = ∫ (sin-1 x)2 dx
Putting sin-1x = t
⇒ x = sin t
dx = cos t dt
∴ I = ∫ (sin-1 x)2 dx
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 11
I = t2 ∫ cos t dt - ∫{\(\frac{d}{d t}\) (t2) ∫ cos t dt } dt
= t2 sin t - ∫ 2t sin t dt
= t2 sin t - 2 ∫ t sin t dt
I = t2 sin t - 2I1 ........ (1)
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 12
= t(- cos t) - ∫1.(- cos t) dt
= - t cos t + ∫cos t dt
= - t cos t + sin t + C1
Putting the value of I1 in equation (1), we have
I = t2 sin t - 2(- t cos t + sin t + C1)
= t2 sin t + 2t cos t - 2 sin t - 2C1
= t2 sin t + 2t cos t - 2 sin t + C (∵ C = - 2C1)
= t2 sin t + 2t \(\sqrt{1-\sin ^{2} t}\) - 2 sin t + C
= x(sin-1 x)2 + 2 sin-1 x\(\sqrt{1-x^{2}}\) - 2x + C

Question 11.
\(\frac{x \cos ^{-1} x}{\sqrt{1-x^{2}}}\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 13
= -[t sin t - ∫1.sin t dt]
= - t sin t + ∫ sin t dt
= - t sin t + (- cos t) + C
= - t sin t - cos t + C
= - t\(\sqrt{1-\cos ^{2} t}\) - cos t + C
= - (cos-1 x)\(\sqrt{1-x^{2}}\) - x + C

Question 12.
x sec2 x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 14
= x tan x - ∫1. tan x dx
= x tan x - ∫tan x dx
= x tan x - (- log |cos x|) + C
= x tan x + log |cos x| + C

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 13.
tan-1 x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 15

Question 14.
x (log x)2
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 16

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 15.
(x2 + 1) log x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 17

Question 16.
ex (sin x + cos x)
Answer:
Let I = ∫ex (sin x + cos x) dx
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 18
= ex - ∫ (cos x) ex dx + ∫ex cos x dx + C
= ex - ∫ex cos x dx + ∫ex cos x dx + C
= ex - ∫ex cos x dx + ∫ex cos x dx + C
= ex sin x + C

Alternative: (i) ∫ex (sin x + cos x) dx
Here f(x) = sin x and f’(x) = cos x
∵ ∫ ex [f(x) + f’(x)] = ex f(x) + C
∴ ∫ex (sin x + cos x) dx = ex sin x + C

(ii) Let I = ∫ex (sin x + cos x) dx
Putting ex sin x = t
⇒ (ex cos x + ex sin x)dx = dt
∴ I = ∫1 dt = t + C = ex sin x + C

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 17.
\(\frac{x e^{x}}{(1+x)^{2}}\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 19

Question 18.
ex \(\left(\frac{1+\sin x}{1+\cos x}\right)\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 20

Question 19.
ex \(\left(\frac{1}{x}-\frac{1}{x^{2}}\right)\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 21

Alternative:
I = ∫ex \(\left(\frac{1}{x}-\frac{1}{x^{2}}\right)\) dx
Here f(x) = \(\frac{1}{x}\) and f'(x) = - \(\frac{1}{x^{2}}\)
From formula
∫ex[f(x) + f'(x)] dx = ex f(x) + C
I = \(\int e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right) d x=e^{x} \frac{1}{x}+C=\frac{e^{x}}{x}+C\)

RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Question 20.
\(\frac{(x-3) e^{x}}{(x-1)^{3}}\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 22

Alternative:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 23

Question 21.
e2x sin x
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 24
I = e2x ∫sin x dx - ∫ {\(\frac{d}{d x}\) e2x ∫ sin x dx } dx
= e2x (- cos x) - ∫2e2x (- cos x) dx
= - e2x cos x + 2∫e2x cos x dx
I = - e2x cos x + 2I1 ........ (1)
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 25
⇒ I = e2x ∫cos x dx - ∫ {\(\frac{d}{d x}\) e2x ∫ cos x dx } dx
⇒ I = e2x sin x - ∫ 2e2x sin x dx + C1
Putting the value of I1 in equation (1), we get
I = - e2x cos x + 2[e2x sin x - 2 ∫e2x sin x dx + C1]
⇒ I = - e2x cos x + 2e2x sin x - 4 ∫e2x sin x dx + 2C1
⇒ I = - e2x cos x + 2e2x sin x - 4I + 2C1
⇒ I + 4I = - e2x cos x + 2e2x sin x + 2C1
⇒ 5I = - e2x cos x + 2e2x sin x + 2C1
⇒ I = - \(\frac{1}{5}\)e2x cos x + \(\frac{2}{5}\)e2x sin x + \(\frac{2}{5}\)C1
⇒ I = - \(\frac{1}{5}\)e2x cos x + \(\frac{2}{5}\)e2x sin x + C
(where C = \(\frac{2}{5}\)C1)
I = \(\frac{e^{2 x}}{5}\) (2 sin x - cos x) + C

Question 22.
sin-1\(\left(\frac{2 x}{1+x^{2}}\right)\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 26
= 2θ tan θ - 2∫1.tan θ dθ
= 2θ tan θ - 2∫tan θ dθ
= 2θ tan θ - 2(- log | cos θ|) + C
= 2θ tan θ + 2 log |cos θ| + C
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 27
RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 28

Question 23.
∫x2 ex3 dx equals:
(A) \(\frac{1}{3}\)ex3 + C
(B) \(\frac{1}{3}\)ex2 + C
(C) \(\frac{1}{2}\)ex3 + C
(D) \(\frac{1}{2}\)ex2 + C
Answer:
Let I = ∫x2 ex3 dx
Putting x3 = t
∴ I = \(\frac{1}{3}\)∫et dt = \(\frac{1}{3}\)et + C
= \(\frac{1}{3}\)ex3 + C
Hence, (A) is the correct answer.

Question 24.
∫ex sec x (1 + tan x) dx equals:
(A) ex cos x + C
(B) ex sec x + C
(C) ex sin x + C
(D) ex tan x + C
Answer:
I = ∫ex sec x (1 + tan x) dx
= ∫ex (sec x + sec x + tan x) dx
Here, f(x) = sec x and f'(x) = sec x tan x
Then, from formula
∫ex [f(x) + f'(x)]dx = ex f(x) + C
I = ex sec x + C
Hence, (B) is the correct answer.

Bhagya
Last Updated on Nov. 3, 2023, 9:31 a.m.
Published Nov. 2, 2023