Evaluate the following integral: `int_0^(pi//2)sqrt(sinvarphi)cos^5varphi dvarphi`

Explanation

Let I=`int_0^(pi//2)sqrt(sinvarphi)cos^5varphi dvarphi`=`int_0^(pi//2)sqrt(sinvarphi)cos^4varphi dvarphi`

Also, let `sin varphi=t⇒cosvarphi dvarphi=dt`

When` varphi =0,t=0 `and when `varphi=π/2​,t=1`

`I=int _0^1 sqrt(​t)
​(1−t^2)^2dt`

`=int _0^1 ​t^(1/2)​(1+t^4−2t^2)dt`

=`int _0^1​[t^(1/2)​+t^(9/2)​−2t^(5/2)​]dt`


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