GSEB Solutions Class 12 Maths Chapter 7 Integrals Ex 7.11

Gujarat Board GSEB Textbook Solutions Class 12 Maths Chapter 7 Integrals Ex 7.11 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 11 Maths Chapter 7 Integrals Ex 7.11

By using properties of definite integrals, evaluate the following:
Question 1.
\(\int_{0}^{\frac{\pi}{2}}\) cos²x
Solution:

Question 2.
\(\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\)dx
Solution:

Question 3.
\(\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}\)dx
Solution:

Question 4.
\(\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{5} x}{\sin ^{5} x+\cos ^{5} x}\)
Solution:

Question 5.
\(\int_{-5}^{5}\)|x + 2|dx
Solution:

Question 6.
\(\int_{2}^{8}\)|x – 5|dx
Solution:

Question 7.
\(\int_{0}^{1}\)x(1 – x)ⁿdx
Solution:

Question 8.
\(\int_{0}^{\frac{\pi}{2}}\)log(1 + tan x)dx
Solution:

Question 9.
\(\int_{0}^{2}\)x\(\sqrt{2-x}\)dx
Solution:

Question 10.
\(\int_{0}^{\frac{\pi}{2}}\)(2log sin x – log sin2x)dx
Solution:

Question 11.
\(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x d x\)
Solution:
Let f(x) = sin²x. Then,
f(- x) = [sin(- x)]² = (- sin x)² = sin²x = f(x).
∴ f(x) is an even function.

Question 12.
\(\int_{0}^{π}\) \(\frac{xdx}{1+sinx}\)
Solution:

Adding (1) and (2), we have:

Question 13.
\(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{7} x d x\)
Solution:
Let f(x) = sin⁷x
f(- x) = [sin (- x)]⁷ = (- sin x)⁷ = – sin⁷x = – f(x)
⇒ f(x) is an odd function of x.
But \(\int_{-a}^{a}\)f(x) dx = 0 when x is odd.
∴ \(\int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{7} x\) dx = 0.

Question 14.
\(\int_{0}^{2π}\) cos⁵x
Solution:
f(x) = cos⁵x
∴ f(2π – x) = cos5(2π – x) = cos⁵x = f(x).
∴ I = \(\int_{0}^{2π}\) cos⁵x dx = 2\(\int_{0}^{π}\)cos⁵x dx.
Again taking g(x) = cos⁵x, we get
g(π – x) = cos⁵(π – x) = – cos⁵x = – g(x)
∴ I = 0. (Because g is an odd function.)
Hence, \(\int_{0}^{2π}\)cos⁵x dx = 0.

Question 15.
\(\int_{0}^{\frac{\pi}{2}}\) \(\frac{sinx-cosx}{1+sinxcosx}\)dx
Solution:

Adding (1) and (2), we get

Question 16.
\(\int_{0}^{π}\)log(1 + cosx)dx
Solution:

Adding (4) and (5), we get

Put 2x = t so that 2 dx = dt.

Question 17.
\(\int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}\)dx
Solution:

Question 18.
\(\int_{0}^{1}\)|x – 1|dx
Solution:

Question 19.
Prove that \(\int_{0}^{a}\) f(x)g(x) dx = 2\(\int_{0}^{a}\) f(x)dx,
if f and g are defined as f(x) = f(a – x) and g(x) + g(x – a) = 4.
Solution:

Hence, the result.

Choose the correct answers in questions 20 and 21:
Question 20.
The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\)(x³ + x cosx + tan⁵x + 1) dx is
(A) 0
(B) 2
(C) π
(D) 1
Solution:

∴ Part(C) is the correct answer.

Question 21.
The value of \(\int_{0}^{\frac{\pi}{2}}\) log (\(\frac{4+3sinx}{4+3cosx}\)) dx is
(A) 2
(B) \(\frac{3}{4}\)
(C) 0
(D) – 2
Solution:

∴ I = 0.
∴ Part(C) is the correct answer.