Evaluate the integral integrate of (sin(x))^2(cos(x))^4 dx

`sin2x =
2sinx*cosx`

`(sin2x)^2 = 4(sinx)^2*(cos(x))^2`

`(sinx)^2*(cos(x))^2 = 1/4(sin2x)^2`

 

`cos2A =
2cos^2A-1 = 1-2sin^A`

`cos^2A = 1/2(cos2A+1)`

`sin^2A =
1/2(1-cos2A)`

 

int(sin(x))^2(cos(x))^4 dx

`= int (sinx)^2*(cosx)^2*(cosx)^2dx`

`= int
(1/4(sin2x)^2)(1/2(cos2x+1))dx`

`= 1/8int [(sin2x)^2cos2x+(sin2x)^2]dx`

`= 1/8[int(sin2x)^2cos2xdx+int1/2(1-cos4x)dx]`

 

 

`int(sin2x)^2cos2xdx`

Let
`t = sin2x`

`dt = 2cos2xdx`

dt/2 = cos2xdx

 

`int(sin2x)^2cos2xdx`

`= int t^2dt/2`

= 1/2(t^3/3)

`= t^3/6`

` =
(sin2x)^3/6`

 

 

`int1/2(1-cos4x)dx`

`= 1/2[int 1dx-intcos4xdx]`

`= 1/2(x-1/4(sin4x))`

`= 1/8(4x-sin4x)`

 

`int(sin(x))^2(cos(x))^4 dx`

`=
1/8[int(sin2x)^2cos2xdx+int(sin2x)^2cos2xdx]`

`=
1/8[(sin2x)^3/6+1/8(4x-sin4x)]+C`

`= 1/24[4(sin2x)^3+3(4x-sin4x)]+C`

C is a constant since this is indefinite integral.

 

`int(sin(x))^2(cos(x))^4 dx = 1/24[4(sin2x)^3+3(4x-sin4x)]+C`