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

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.