Mathematics

# Evaluate $\displaystyle\int_0^\infty{f(x^n+x^{-n})\ln{x}\frac{dx}{x}}$

$0$

##### SOLUTION
Let $t=\ln{x}\implies x=e^t$
$dx=e^tdt$ or $\displaystyle\frac{dx}{x}=dt$
Also, $x=0\implies t=-\infty$
and $x=\infty\implies t=\infty$
Thus, $\displaystyle\int_0^\infty{f(x^n+x^{-n})\ln{x}\frac{dx}{x}}=\int_{-\infty}^\infty{f(e^{nt}+e^{-nt}).tdt}=0$

Ans: A

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 One Word Medium
$\displaystyle \int \frac{e^{2x}dx}{\sqrt[4]{\left ( e^{x}+1 \right )}}=\frac{k}{21}\left ( e^{x}+1 \right )^{3/4}\left [ 3e^{x}-4 \right ].$ Find the value of $k$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \dfrac{\sin x \cos x }{\sqrt{1 - \sin^4 x}}dx$  is equal to
• A. $\dfrac{1}{2} (\sin^2 x) + C$
• B. $\tan^{-1} (\sin^2 x) + C$
• C. $\cos^{-1}(\sin x) + c$
• D. $\dfrac{1}{2} \sin^{-1}(\sin^2 x) + C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\int { \sin ^{ 2 }{ \left( 2x+5 \right) } } dx\quad$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve: $\displaystyle \overset{\beta}{\underset{\alpha}{\int}} \sqrt{(x - \alpha)(\beta - x)}dx$.

$\int(2x^2-3 \, sin x+5 \sqrt{x})dx$