Mathematics

Solve:$$\displaystyle \int_0^{\pi/2}sinx \cos^3 xdx$$


SOLUTION
$$\int _{ 0 }^{ \dfrac { \pi  }{ 2 }  }{ \sin x{ \cos }^{ 3 }xdx } \\ Let\quad \cos x=t\\ \Rightarrow \sin xdx=-dt\\ also,\quad at\quad x=0;t=1\\ and\quad at\quad x=\dfrac { \pi  }{ 2 } ;t=0\\ \therefore \int _{ 0 }^{ \dfrac { \pi  }{ 2 }  }{ \sin x{ \cos }^{ 3 }xdx } =-\int _{ 1 }^{ 0 }{ { t }^{ 3 }dt } \\ ={ -\left[ \dfrac { { t }^{ 4 } }{ 4 }  \right]  }_{ 1 }^{ 0 }\\ =\dfrac { 1 }{ 4 } $$
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Subjective Medium Published on 17th 09, 2020
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