NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3

These NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3 Questions and Answers are prepared by our highly skilled subject experts. https://mcq-questions.com/ncert-solutions-for-class-12-maths-chapter-7-ex-7-3/

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.3

Ncert Solutions Class 12 Maths Chapter 7 Exercise 7.3 Question 1.
sin²(2x+5)
Solution:
∫sin²(2x+5)dx
= \(\frac { 1 }{ 2 }\)∫[1-cos2(2x+5)]dx
= \(\frac { 1 }{ 2 }\)∫[1-cos(4x+10)]dx
= \(\frac { 1 }{ 2 } \left[ x-\frac { sin(4x+10) }{ 4 } \right]\) + C

Question 2.
sin3x cos4x
Solution:
∫sin3x cos4x
= \(\frac { 1 }{ 2 }\)∫[sin(3x+4x)+cos(3x-4x)]dx
= \(\frac { 1 }{ 2 }\)∫[sin7x+sin(-x)]dx
= \(-\frac { 1 }{ 14 } cos7x+\frac { 1 }{ 2 } cosx\) + C

Question 3.
cos 2x cos 4x cos 6x dx
Solution:

Question 4.
sin³(2x + 1)dx
Solution:

Question 5.
sin³ x cos³ x
Solution:

Question 6.
sin x sin 2x sin 3x
Solution:

Question 7.
sin 4x sin 8x
Solution:
∫sin 4x sin 8x dx = \(\frac { 1 }{ 2 }\)∫sin 4x sin 8xdx
= \(\frac { 1 }{ 2 }\)∫(cos 4x – cos 12x)dx
= \(\frac { 1 }{ 2 } \left[ \frac { sin4x }{ 4 } -\frac { sin12x }{ 12 } \right] \) + C

Question 8.
\(\frac { 1-cosx }{ 1+cosx }\)
Solution:
\(\int { \frac { 1-cosx }{ 1+cosx } dx } \)
\(\int { \frac { { 2sin }^{ 2 }\frac { x }{ 2 } }{ { 2cos }^{ 2 }\frac { x }{ 2 } } dx } =\int { { tan }^{ 2 }\frac { x }{ 2 } dx } \)
\(=\int { \left[ { sec }^{ 2 }\frac { x }{ 2 } -1 \right] } dx\quad =2tan\frac { x }{ 2 } \) – x + C

Question 9.
\(\frac { cosx }{ 1+cosx }\)
Solution:

Question 10.
∫sinx⁴ dx
Solution:

Question 11.
cos⁴ 2x
Solution:

Question 12.
\(\frac { { sin }^{ 2 }x }{ 1+cosx } \)
Solution:

Question 13.
\(\frac { cos2x-cos2\alpha }{ cosx-cos\alpha } \)
Solution:

Question 14.
\(\frac { cosx-sinx }{ 1+sin2x }\)
Solution:

Question 15.
tan³ 2x sec 2x
Solution:

Question 16.
tan⁴x
Solution:

Question 17.
\(\frac { { sin }^{ 3 }x+{ cos }^{ 3 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } \)
Solution:

Question 18.
\(\frac { cos2x+{ 2sin }^{ 2 }x }{ { cos }^{ 2 }x } \)
Solution:

Question 19.
\(\frac { 1 }{ sinx{ cos }^{ 3 }x } \)
Solution:

Question 20.
\(\frac { cos2x }{ { (cosx+sinx) }^{ 2 } } \)
Solution:

Question 21.
sin-1 (cos x)
Solution:
\(\int \sin ^{-1}(\cos x) d x=\int \sin ^{-1}\left[\sin \left(\frac{\pi}{2}-x\right)\right] \) dx
\(\int\left(\frac{\pi}{2}-x\right) d x=\frac{\pi}{2} x-\frac{x^{2}}{2}\) + C

Question 22.
\(\int { \frac { 1 }{ cos(x-a)cos(x-b) } dx } \)
Solution:

Question 23.
\(\int { \frac { { sin }^{ 2 }x-{ cos }^{ 2 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } } dx\quad is\quad equal\quad to\)
(a) tan x + cot x + C
(b) tan x + cosec x + C
(c) -tan x + cot x +C
(d) tan x  + sec x + C
Solution:
(a) \(\int { \frac { { sin }^{ 2 }x-{ cos }^{ 2 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } } \) dx
= \(\int \frac{\sin ^{2} x}{\sin ^{2} x \cos ^{2} x} d x-\int \frac{\cos ^{2} x}{\sin ^{2} x \cos ^{2} x} d x\)
= ∫(sec² x dx ∫cosec² x dx
= tan x + cot x + C

Question 24.
\(\int { \frac { e^{ x }(1+x) }{ cos^{ 2 }({ e }^{ x }.{ x }) } } dx\quad is\quad equal\quad to\)
(a) – cot(e.x^x) + C
(b) tan(xe^x) + C
(c) tan(e^x) + C
(d) cot e^x + C
Solution:
(b) \(\int { \frac { e^{ x }(1+x) }{ cos^{ 2 }({ e }^{ x }.{ x }) } } dx\)
= ∫sec²t dt
= tan t+c = tan(xe^x) + C