Mathematics

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


ANSWER

$$0$$


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
View Full Answer

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

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 $$k$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$ \dfrac{1}{2} \sin^{-1}(\sin^2 x) + C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Easy
$$\int(2x^2-3 \, sin x+5 \sqrt{x})dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer