Mathematics

$$\int_{0}^{\pi/2}\dfrac {\sin x \cos x}{1+\sin^{4}x}dx$$


SOLUTION
$$\begin{array}{l} \int _{  }^{  }{ \frac { { \cos  \left( x \right) \sin  \left( x \right)  } }{ { { { \sin   }^{ 4 } }\left( x \right) +1 } } dx }  \\ Substitute\, u={ \sin ^{ 2 }  }x,dx=\frac { 1 }{ { 2\cos  x\sin  x } }du  \\ =\frac { 1 }{ 2 } \int _{  }^{  }{ \frac { 1 }{ { 1+{ u^{ 2 } } } } du }  \\ =\frac { 1 }{ 2 } { \tan ^{ -1 }  }u+C \end{array}$$

Apply the limit, we get
$$\int_0^{\frac{\pi }{2}} {\frac{{\cos \left( x \right)\sin \left( x \right)}}{{{{\sin }^4}\left( x \right) + 1}}dx = \frac{\pi }{8}} $$
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Subjective Medium Published on 17th 09, 2020
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