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}^3x4 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}}tan2 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}} cos2 2x sin3 4x dx
Solution:
Question 3.
(i) \int_1^2e4x+1 dx
Solution:
\int_1^2e4x+1 dx = \left[\frac{e^{4 x+1}}{4}\right]_1^2 = \frac{1}{4}{e9 – e5}
(ii) \int_0^23x+2 dx
Solution:
\int_0^23x+2 dx = \left[\frac{3^{x+2}}{\ln 3}\right]_0^2 = \frac{1}{\ln 3}{34 – 32} = \frac{72}{\ln 3}
(iii) \int_0^1cos 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[x2] dx
Solution:
Question 7.
(i) \int_1^2ex . (x + 1) dx
Solution:
(ii) \int_0^{\frac{\pi}{4}}x sin x dx
Solution:
(iii) \int_1^2x log x dx
Solution:
(iv) \int_0^1x 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}}ex cos x dx
Solution: