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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