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.