GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.3

Gujarat Board GSEB Textbook Solutions Class 12 Maths Chapter 7 Integrals Ex 7.3 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 11 Maths Chapter 7 Integrals Ex 7.3

Find the integrals of the following:
Question 1.
sin²(2x + 5)
Solution:

Question 2.
sin3x cos4x
Solution:

Question 3.
cos2x cos4x cos6x
Solution:

Question 4.
sin³(2x + 1)
Solution:

Question 5.
sin³x cos³x
Solution:

Question 6.
sinx sin2x sin3x
Solution:

Question 7.
sin4x sin8x
Solution:

Question 8.
\(\frac{1-cosx}{1+cosx}\)
Solution:

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

Question 10.
sin⁴x
Solution:

Question 11.
cos⁴2x
Solution:

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

= x – sinx + C.

Question 13.
\(\frac{cos2x-cos2α}{cosx-cosα}\)
Solution:

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

Question 15.
tan³2x sec2x
Solution:

Question 16.
tan⁴x
Solution:

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

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

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

Question 20.
\(\frac{\cos 2 x}{(\cos x+\sin x)^{2}}\)
Solution:

Question 21.
sin^-1x
Solution:

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

Choose the correct answers in the following questions 23 and 24:
Question 23.
∫ \(\frac{\sin ^{2} x-\cos ^{2} x}{\sin ^{2} x \cos ^{2} x}\)
(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:

∴ part (A) is the correct answer.

Question 24.
∫ \(\frac{e^{x}(1+x)}{\cos ^{2}\left(x e^{x}\right)}\) is equal to.
(A) – cot (e.x^x)
(B) tan (x.e^x) + C
(C) tan (e^x) + C
(D) cot (e^x) + C
Solution:
I = ∫ \(\frac{e^{x}(1+x)}{\cos ^{2}\left(x e^{x}\right)}\) dx.
Put x e^x = t, so that (e^x + x e^x) dx = dt.
or e^x(1 + x)dx = dt
∴ I = ∫ \(\frac{d t}{\cos ^{2} t}\) = ∫sec² t dt = tan t + C.
= tan(x e^x) + C.
∴ Part (B) is the correct answer.