Mathematics

# $\int\limits_{\dfrac{{ - \pi }}{4}}^{\dfrac{\pi }{4}} {\dfrac{{{e^x}\left( {x\sin x} \right)}}{{{e^{2x}} - 1}}} \,dx$ is equal to

$0$

##### SOLUTION
Solution
$\displaystyle \int_{-\dfrac{\pi}{4}}^{\dfrac{\pi}{4}} \dfrac{e^{x}(x \sin x )}{e^{2x}-1} dx.$

let $f(x)=\dfrac{e^x (x \sin x)}{e^{2x}-1}$   $f(-x)=\dfrac{e^{-x}(-x\sin - x)}{\dfrac{1-e^{2x}}{e^{2x}}}$

$f(-x)=-\dfrac{e^{x}(x \sin x)}{e^{2x}-1}=-f(x)$

$f(x)$ is an odd function

$\displaystyle \int _{-\dfrac{\pi}{4}}^{\dfrac{\pi}{4}} f(x)=0$ property of define integral.
$A$ is correct

Its FREE, you're just one step away

Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
Evaluate: $\displaystyle \int_{0}^{1} \cos$ $\left(2 \cot^{-1}\sqrt{\displaystyle \frac{1- {x}}{1+ {x}}}\right)dx$
• A. $\dfrac{1}{2}$
• B. $0$
• C. $1$
• D. $\dfrac{-1}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\int_{}^{} {\frac{{dx}}{{x\left( {{x^n} + 1} \right)}}}$ is equal to
• A. $\frac{1}{n}\log \left( {\frac{{{x^n}}}{{{x^n} + 1}}} \right) + c$
• B. $\log \left( {\frac{{{x^n}}}{{{x^n} + 1}}} \right) + c$
• C. None of these
• D. $-\frac{1}{n}\log \left( {\frac{{{x^n} + 1}}{{{x^n}}}} \right) + c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate $\int {{e^x}\left[ {\tan x - \log \left( {\cos x} \right)} \right]} dx$
• A. ${e^x}\log \left( {{\mathop{\rm cosec x}\nolimits} } \right) + C$
• B. ${e^x}\log \left( {\cos x} \right) + C$
• C. ${e^x}\log \left( {\sin x} \right) + C$
• D. ${e^x}\log \left( {\sec x} \right) + C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve : $\underset{0}{\overset{\pi / 4}{\int}} 2 \sin \, x \, \sin 2 x \, dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$I=\displaystyle \int \dfrac{(x+a)^3}{x^3}dx$ is equal to:
• A. $x^2+ 3a \log x -\dfrac{3a^2}{x} - \dfrac{a^3}{2x^2}+c$
• B. $x^3+ 3a \log x -\dfrac{2a^2}{x} - \dfrac{3a^3}{2x^2}+c$
• C. $1+ 2a \log x -\dfrac{2a^2}{x} - \dfrac{3a^2}{2x^2}+c$
• D. $x+ 3a \log x -\dfrac{3a^2}{x} - \dfrac{a^3}{2x^2}+c$