Mathematics

# $\displaystyle \int \left(\frac{x}{1+x^5}\right)^\frac{3}{2}dx$ equals-

$\dfrac{2}{5}\sqrt{\dfrac{x^5}{1+x^5}}+c$

##### SOLUTION
$LHS=\displaystyle\int \left(\dfrac{x}{1+x^{5}}\right)^{3/2}dx$
This can be re-written as
$\displaystyle\int \dfrac{x^{4}}{(1+x^{5})^{2}}\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}dx$
$\dfrac{2}{5}\displaystyle\int \dfrac{1}{2}\dfrac{5x^{4}}{(1+x^{5})^{2}}\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}dx$
$\dfrac{2}{5}\displaystyle\int \dfrac{5x^{4}+5x^{9}-5x^{9}}{(1+x^{5})^{2}}\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}dx$
$\dfrac{2}{5}\displaystyle\int \dfrac{1}{2}\left[\dfrac{5x^{4}(1+x^{5})-5x^{4}(x^{5})}{(1+x^{5})^{2}}\right]\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}dx$
This is $d\left(\dfrac{x^{5}}{1+x^{5}}\right)$
$\Rightarrow \dfrac{2}{5}\displaystyle\int \dfrac{1}{2}\left(\dfrac{1+x^{5}}{x^{5}}\right)^{1/2}d\left(\dfrac{x^{5}}{1+x^{5}}\right)$
let $\dfrac{x^{5}}{1+x^{5}}=t$
then $\dfrac{2}{5}\displaystyle\int\dfrac{1}{2}\left(\dfrac{1}{t}\right)^{1/2}dt$
$=\dfrac{2}{5}\displaystyle\int \dfrac{1}{2} t^{-1/2}dt$
$=\dfrac{2}{5}t^{1/2}+C=\dfrac{2}{5}\sqrt{\dfrac{x^{5}}{1+x^{5}}}+C$

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

#### Realted Questions

Q1 Subjective Medium
Solve
$\int {{x^{ - 3}}\left( {x + 1} \right)dx}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Show that $\displaystyle \int _\limits{0}^{\frac{\pi}{2}}\frac{x}{\sin x +\cos x}dx =\frac{\pi}{2\sqrt2}\log (\sqrt2+1)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Solve: $\displaystyle\int \dfrac{dx}{3+2 \sin x + \cos x}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
The integral $\int \frac{\sin^2x\cos^2x}{\left ( \sin^5x+\cos^3x\sin^2x+\sin^3x\cos^2x+\cos^5x \right )^2}dx$ is equal to
• A. $\frac{1}{3\left ( 1+\cot^3x \right )}+C$
• B. $\frac{-1}{3\left ( 1+\cot^3x \right )}+C$
• C. $\frac{1}{3\left ( 1+\tan^3x \right )}+C$
• D. $\frac{-1}{3\left ( 1+\tan^3x \right )}+C$

Integrate the function   $\displaystyle \frac {1}{x-\sqrt x}$