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

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

Evaluate the following Integrals.

Question 1.

(i) \(\int_{-2}^3\)x^{4} dx

Solution:

(ii) \(\int_0^1 x^{\frac{1}{3}}\) dx

Solution:

\(\int_0^1 x^{\frac{1}{3}}\) dx = \(\left[\frac{x^{\frac{4}{3}}}{\frac{4}{3}}\right]_0^1\) = \frac{3}{4}\(\)

(iii) \(\int_1^4 \frac{d x}{\sqrt{x}}\) dx

Solution:

(iv) \(\int_1^3 \frac{d x}{x^3}\) dx

Solution:

(v) \(\int_1^2\left(4 x+\sqrt{x}+\frac{1}{x^{\frac{1}{3}}}+\frac{1}{x^2}\right)\) dx

Solution:

(vi) \(\int_{-1}^2\)(2x + 1)(x – 2) dx

Solution:

(vii) \(\int_0^1\)(x + 1)^{4} dx

Solution:

(viii) \(\int_0^1 x^7\left(1+x^8\right)^{\frac{1}{3}}\) dx

Solution:

(ix) \(\int_1^4 \frac{x^2-3 x+5}{\sqrt{x}}\) dx

Solution:

(x) \(\int_1^4 \frac{(x+2)\left(x^2+3\right)}{\sqrt{x}}\) dx

Solution:

Question 2.

(i) \(\int_0^{\frac{\pi}{2}}\)(cos x – sin x) dx

Solution:

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

Solution:

(iii) \(\int_0^{\frac{\pi}{4}}\)tan^{2} x dx

Solution:

(iv) \(\int_0^\pi 3 \sin \frac{x}{2} \cos \frac{x}{2}\) dx

Solution:

(v) \(\int_0^{\frac{\pi}{4}}\)sin 2x cos x dx

Solution:

(vi) \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1+\cos 2 x}{1-\cos 2 x}\) dx

Solution:

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

Solution:

(viii) \(\int_0^{\frac{\pi}{4}}\) cos^{2} 2x sin^{3} 4x dx

Solution:

Question 3.

(i) \(\int_1^2\)e^{4x+1} dx

Solution:

\(\int_1^2\)e^{4x+1} dx = \(\left[\frac{e^{4 x+1}}{4}\right]_1^2\) = \(\frac{1}{4}\){e^{9} – e^{5}}

(ii) \(\int_0^2\)3^{x+2} dx

Solution:

\(\int_0^2\)3^{x+2} dx = \(\left[\frac{3^{x+2}}{\ln 3}\right]_0^2\) = \(\frac{1}{\ln 3}\){3^{4} – 3^{2}} = \(\frac{72}{\ln 3}\)

(iii) \(\int_0^1\)cos h 2x dx

Solution:

(iv) \(\int_0^1 \frac{e^x-e^{-x}}{e^x+e^{-x}}\) dx

Solution:

(v) \(\int_0^2 x^2 e^{x^3}\) dx

Solution:

Question 4.

(i) \(\int_0^1 \frac{d x}{\sqrt{1-x^2}}\)

Solution:

(ii) \(\int_{\frac{2}{\sqrt{3}}}^2 \frac{d x}{x \sqrt{x^2-1}}\)

Solution:

(iii) \(\int_0^{\sqrt{3}} \frac{d x}{1+x^2}\)

Solution:

(iv) \(\int_0^{\ln 2} \frac{d x}{\sqrt{e^{2 x}-1}}\) dx

Solution:

Question 5.

(i) \(\int_0^1 \frac{d x}{3 x+2}\)

Solution:

(ii) \(\int_0^4 \frac{d x}{\sqrt{x^2+9}}\)

Solution:

(iii) \(\int_2^3 \frac{d x}{\sqrt{x^2-2}}\)

Solution:

(iv) \(\int_0^4 \sqrt{x^2+9}\) dx

Solution:

Question 6.

(i) \(\int_{-2}^1\)(|x| + x) dx

Solution:

(ii) \(\int_2^5\)[x] dx

Solution:

(iii) \(\int_0^{\frac{3}{2}}\)[2x] dx

Solution:

(iv) \(\int_0^2\)[x^{2}] dx

Solution:

Question 7.

(i) \(\int_1^2\)e^{x} . (x + 1) dx

Solution:

(ii) \(\int_0^{\frac{\pi}{4}}\)x sin x dx

Solution:

(iii) \(\int_1^2\)x log x dx

Solution:

(iv) \(\int_0^1\)x tan^{-1} x dx

Solution:

(v) \(\int_0^1 \frac{x d x}{(2x+1)(x+1)}\)

Solution:

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

Solution:

(vii) \(\int_\theta^{\frac{\pi}{2}}\)e^{x} cos x dx

Solution: