(\integral \_ { 0 } ^ { \pi / 2 } \sin ^ { 2 } x \cos ^ { 3... | MATH - SUMMATION OF SERIES BY DEFINITE INTEGRATION, GAMMA FUNCTION, LEIBNITZ'S RULE | SolveeIt

Question

$\int _ { 0 } ^ { \pi / 2 } \sin ^ { 2 } x \cos ^ { 3 } x d x =$

$\frac { 2 } { 15 }$

$\frac { 4 } { 15 }$

None of these

Answer

$\frac { 2 } { 15 }$

Explanation

Solution

Using gamma function,

$\int _ { 0 } ^ { \pi / 2 } \sin ^ { 2 } x \cos ^ { 3 } x d x = \frac { \Gamma \left( \frac { 3 } { 2 } \right) \Gamma 2 } { 2 \Gamma \left( \frac { 7 } { 2 } \right) } = \frac { 2 } { 15 }$ .

Question: \(\int _ { 0 } ^ { \pi / 2 } \sin ^ { 2 } x \cos ^ { 3 } x d x =\)...

Solution