<s>[INST] Need help with integral and change the variable I have
$$\displaystyle \int_0^\pi \sqrt{x} ~\cos x ~dx$$
and I need to make change of the variable $u = \sin x$.
 [/INST] Hint: Try this if you want:
$$x\sqrt { x } =u\\ x={ u }^{ \frac { 2 }{ 3 }  }\\ \sqrt { x } ={ u }^{ \frac { 1 }{ 3 }  }\\ dx=\frac { 2 }{ 3 } { u }^{ \frac { -1 }{ 3 }  }du\\ \int _{ 0 }^{ \pi  } \sqrt { x } \cos  x~ dx=\int _{ 0 }^{ \pi  }{ u } ^{ \frac { 1 }{ 3 }  }\cos { u } ^{ \frac { 2 }{ 3 }  }\frac { 2 }{ 3 } { u }^{ \frac { -1 }{ 3 }  }du=\frac { 2 }{ 3 } \int _{ 0 }^{ \pi\sqrt\pi  }{ \cos { u } ^{ \frac { 2 }{ 3 }  }du } $$
$\int { \cos { u } ^{ \frac { 2 }{ 3 }  }du } $ does not have an analytical expression. You can try on wolframalpha.com .
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