Mathematics

Evaluate the following definite integrals:$\displaystyle \int _{\pi /6}^{\pi /4} cosec \ x \ dx$

SOLUTION
$I=\displaystyle \int _{\pi /6}^{\pi /4} cosec \ x \ dx$

$=[\log (cosec \ x-\cot x)]_{\pi /6}^{\pi /4}$

$=\log \left (cosec \dfrac {\pi}{4} -\cot \dfrac {\pi}{4}\right) -\log \left (cosec \dfrac {\pi}{6}-\cot \dfrac {\pi}{6} \right)$

$=\log (\sqrt 2-1)-\log (2-\sqrt 3)$

$=\log \left (\dfrac {\sqrt 2-1}{2-\sqrt 3}\right )$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

Realted Questions

Q1 Single Correct Medium
$\displaystyle \int {{e^x}\dfrac{{x - 1}}{{{{\left( {x + 1} \right)}^3}}}{\text{dx}}\;{\text{equals}}}$
• A. $- \dfrac{{{e^x}}}{{x + 1}} + c$
• B. $\dfrac{{{e^x}}}{{x + 1}} + c$
• C. $- \dfrac{{{e^x}}}{{{{(x + 1)}^2}}}$
• D. $\dfrac{{{e^x}}}{{{{(x + 1)}^2}}} + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle \int \sqrt{\frac{\cos x-\cos ^{3}x}{1-\cos ^{3}x}}dx$ is equal to
• A. $\displaystyle \frac{2}{3}\sin ^{-1}(\cos ^{3/2}x)+C$
• B. $\displaystyle \frac{3}{2}\sin ^{-1}(\cos ^{3/2}x)+C$
• C. none of these
• D. $\displaystyle \frac{2}{3}\cos ^{-1}(\cos ^{3/2}x)+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the definite integral:
$\displaystyle \int_{0}^{\pi /2} \cos x\ dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\int\limits_0^1 x \sqrt {\frac{{1 - {x^2}}}{{1 + {x^2}}}dx}$
• A. $\frac{\pi }{4}$
• B. $\frac{1}{2}$
• C. $\frac{\pi }{4} + \frac{1}{2}$
• D. $\frac{\pi }{4} - \frac{1}{2}$

Solve $\displaystyle \int \cos^{3}x\ dx$