Mathematics

Evaluate $\int\limits_0^{\dfrac{\pi }{2}} {{{\sin }^4}x{{\cos }^5}xdx.}$

SOLUTION

Consider the given integral.

$I=\int_{0}^{\dfrac{\pi }{2}}{{{\sin }^{4}}x{{\cos }^{5}}x}dx$

$I=\int_{0}^{\dfrac{\pi }{2}}{{{\sin }^{4}}x\cos x}{{\left( {{\cos }^{2}}x \right)}^{2}}dx$

$I=\int_{0}^{\dfrac{\pi }{2}}{{{\sin }^{4}}x\cos x}{{\left( 1-{{\sin }^{2}}x \right)}^{2}}dx$

Let $t=\sin x$

$\dfrac{dt}{dx}=\cos x$

$dt=\cos dx$

Therefore,

$I=\int_{0}^{1}{{{t}^{4}}}{{\left( 1-{{t}^{2}} \right)}^{2}}dt$

$I=\int_{0}^{1}{{{t}^{4}}}\left( 1+{{t}^{4}}-2{{t}^{2}} \right)dt$

$I=\int_{0}^{1}{\left( {{t}^{4}}+{{t}^{8}}-2{{t}^{6}} \right)}dt$

$I=\left[ \dfrac{{{t}^{5}}}{5}+\dfrac{{{t}^{9}}}{9}-\dfrac{2{{t}^{7}}}{7} \right]_{0}^{1}$

$I=\left[ \left( \dfrac{{{1}^{5}}}{5}+\dfrac{{{1}^{9}}}{9}-\dfrac{2{{\left( 1 \right)}^{7}}}{7} \right)-\left( 0 \right) \right]$

$I=\dfrac{1}{5}+\dfrac{1}{9}-\dfrac{2}{7}$

$I=\dfrac{63+35-90}{315}$

$I=\dfrac{8}{315}$

Hence, this is the answer.

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Subjective Medium Published on 17th 09, 2020
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