Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Ex 9(k) Textbook Exercise questions and Answers.

## CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Exercise 9(k)

Evaluate the following Integrals:

Question 1.

(i) \(\int_0^{\frac{\pi}{2}} \frac{d x}{1+\tan x}\)dx

Solution:

(ii) \(\int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\)dx

Solution:

(iii) \(\int_0^1 \frac{\ln (1+x)}{2+x^2}\)dx (x = tan θ)

Solution:

(iv) \(\int_0^\pi \frac{x d x}{1+\sin x}\)

Solution:

Question 2.

(i) \(\int_{-a}^a\)x^{4} dx

Solution:

(ii) \(\int_{-a}^a\)(x^{5} + 2x^{2} + x) dx

Solution:

(iii) \(\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\)cos^{2} x dx

Solution:

(iv) \(\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\)sin^{5} x dx

Solution:

Let f(x) = sin^{5} x

Then f(-x) = sin^{5} (-x)

= -sin^{5} x = -f(x)

So f(x) is an odd function.

Thus \(\int_{-a}^a\)f(x) dx = 0

\(\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\)sin^{5} x dx = 0

Question 3.

(i) \(\int_0^\pi\)cos^{3} x dx

Solution:

(ii) \(\int_0^\pi\)cos^{2} x dx

Solution:

(iii) \(\int_0^\pi\)sin^{3} x cos x dx

Solution:

\(\int_0^\pi\)sin^{3} x cos x dx

[Put sin x = t, then cos x dx = dt

When x = 0, t = 0, when x = π, t = 0

\(\int_0^\pi\)t^{3} dt = 0

(iv) \(\int_0^\pi\)sin x cos^{2} x dx

Solution:

Question 4.

Show that

(i) \(\int_0^1 \frac{\ln x}{\sqrt{1-x^2}}\) dx = \(\frac{\pi}{2} \ln \frac{1}{2}\)

Solution:

(ii) \(\int_0^{\frac{\pi}{2}} \frac{\cos x-\sin x}{1+\sin x \cos x}\) dx = 0

Solution:

(iii) \(\int_0^\pi\)x ln sin x dx = \(\frac{\pi^2}{2} \ln \frac{1}{2}\)

Solution:

Question 5.

(i) \(\int_0^{\pi / 2}\)ln (tan x + cot x) dx

Solution:

(ii) \(\int_0^\pi \frac{x \tan x-\sin x}{1+\sin x \cos x}\) dx

Solution:

(iii) \(\int_1^3 \frac{\sqrt{x} d x}{\sqrt{4-x}+\sqrt{x}}\)

Solution:

(iv) \(\int_0^\pi \frac{x \sin x d x}{1+\cos ^2 x}\)

Solution:

(v) \(\int_0^1\)x (1 – x)^{100} dx

Solution:

(vi) \(\int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\cot x}}\)

Solution:

(vii) \(\int_0^{50}\)e^{x-[x]} dx

Solution: