Mathematics

# The value of  $\displaystyle\int_{0}^{\dfrac{\pi}{2}}{\dfrac{dx}{1+{\tan}^{3}{x}}}$ is

##### SOLUTION
$I=\displaystyle\int_{0}^{\dfrac{\pi}{2}}{\dfrac{dx}{1+{\tan}^{3}{x}}}$          ......$(1)$

Replace $x\rightarrow \dfrac{\pi}{2} - x$

$I=\displaystyle\int_{0}^{\dfrac{\pi}{2}}{\dfrac{dx}{1+{\tan}^{3}{\left(\dfrac{\pi}{2}-x\right)}}}$

$=\displaystyle\int_{0}^{\dfrac{\pi}{2}}{\dfrac{dx}{1+{\cot}^{3}{x}}}$

$=\displaystyle\int_{0}^{\dfrac{\pi}{2}}{\dfrac{dx}{1+\dfrac{1}{{\tan}^{3}{x}}}}$

$=\displaystyle\int_{0}^{\dfrac{\pi}{2}}{\dfrac{{\tan}^{3}{x}dx}{{\tan}^{3}{x}+1}}$        ......$(2)$

Adding $(1)$ and $(2)$ we get

$\Rightarrow 2I =\displaystyle\int_{0}^{\dfrac{\pi}{2}}{\dfrac{1+{\tan}^{3}{x}}{1+{\tan}^{3}{x}}dx}$

$\Rightarrow 2I=\displaystyle\int_{0}^{\dfrac{\pi}{2}}{dx}$

$\Rightarrow 2I=\left[x\right]_{0}^{\dfrac{\pi}{2}}$

$\Rightarrow 2I=\dfrac{\pi}{2}$

$\therefore I=\dfrac{\pi}{4}$

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

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