Mathematics

Evaluate $$\int { \dfrac { \tan ^{ 7 }{ \sqrt { x }  } \sec ^{ 2 }{ \sqrt { x }  }  }{ \sqrt { x }  }  } dx$$


SOLUTION
$$\int_{}^{} {\cfrac{{{{\tan }^7}\sqrt x {{\sec }^2}\sqrt x }}{{\sqrt x }}dx} $$
putting $$\tan \sqrt x  = t$$
$$ \Rightarrow {\sec ^2}\sqrt x  \times \cfrac{1}{{2\sqrt x }}dx = dt$$
$$ \Rightarrow \cfrac{{{{\sec }^2}\sqrt x }}{{\sqrt x }}dx = 2dt$$
$$ = 2\int_{}^{} {{t^7}dt} $$
$$ = 2 \times \cfrac{{{t^8}}}{8} + c$$
$$ = \cfrac{1}{4}{\tan ^8}\sqrt x  + c$$
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Subjective Medium Published on 17th 09, 2020
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