Mathematics

# Solve: $\displaystyle\int_{0}^{1}\dfrac{1-x^{2}}{(1+x^{2})^{2}}dx$

##### SOLUTION

$I=\displaystyle\int_{0}^{1}\dfrac{x^{\frac{1}{2}}-1}{\left(x+\dfrac{1}{x}\right)^{2}}\ dx$

Let $x+\dfrac{1}{x}=t$.

Then, $d=\left(x+\dfrac{1}{x}\right)=dt$

$\Rightarrow \left(1-\dfrac{1}{x^{2}}\right)dx=dt$

Clearly,

$x=0\Rightarrow t=\infty$

$x=1\Rightarrow t=2$

$\therefore\quad =\displaystyle\int_{\infty}^{2}-\dfrac{1}{t^{2}}dt$

$=\left[\dfrac{1}{t}\right]_{\infty}^{2}$   [$\because\int x^n=\dfrac{x^{n+1}}{n+1}$]

$=\dfrac{1}{2}$

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

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