`int_(0)^(pi//2) sin^(4)xdx` का मान ज्ञात कीजिए

Explanation

`underset(0)overset(pi//2)int sin^(4)xdx`
`=underset(0)overset(pi//2)int (sin^(2))^(2)dx`
`=1/4 underset(0)overset(pi//2)int (2sin^(2)x)^(2)dx`
`=1/4 underset(0)overset(pi//2)int (1-cos 2x)dx`
`=1/4 underset(0)overset(pi//2)int [1+cos2x-2 cos 2x]dx`
`=1/4 underset(0)overset(pi//2)int [1+((1+cos 4x)/(2))-2cos2x]dx`
`=1/4 underset(0)overset(pi//2)int [1-2cos 2x+1/2+(cos 4x)/(2)]dx`
`=1/4 underset(0)overset(pi//2)int [3/2-2cos 2x+(cos 4x)/(2)]dx`
`=1/8 underset(0)overset(pi//2)int [3-4cos 2x+cos 4x]dx`
`=1/8 [3x-4((sin 2x)/(2))+((sin 4x)/(4))]_(0)^(pi//2)`
`=1/8[3x-2sin2x+1/4sin 4x]_(0)^(pi//2)` `-{3 xx 0-2 sin 0^(@)+1/4 sin 60^(@)}]`
`=1/8[{(3pi)/(2)-0+1/4xx0}-0]=(3pi)/(16)`