`int (sin^3(sqrt(x)))/sqrt(x) dx` Evaluate the integral

`int[(sin^3sqrtx)/(sqrtx)]dx=`

Integrate using the u-substitution method. For this problem the u-substitution method will be used twice. The first time we substitute, let's use the variable y.

Let

`y=sqrtx`

`(dy)/dx=1/(2sqrtx)`

`dx=2sqrtxdy`

`int[(sin^3(y))/y*2sqrtxdy=`

`2int[(sin^3(y))/(y)]*ydy=`

`2intsin^3(y)dy=`

`2intsin^2(y)sin(y)dy=`

`2int(1-cos^2(y))sin(y)dy=`

The u-substitution method will be used a second time. We will use the variable u.

Let

`u=cosy`

`(du)/dy=-sin(y)`

`dy=(-sin(y))/(du)`

`2int(1-u^2)sin(y)[(du)/(-sin(y))]=`

`-2int(1-u^2)du=`

`-2[u-1/3u^3]+C=`

`-2u+2/3u^3+C`

Substitute in for u. `u=cos(y)`

`-2cos(y)+2/3cos^3(y)+C` 

Substitute in for y. `y=sqrtx`

`-2cos(sqrtx)+2/3cos^3(sqrtx)+C`

The final answer is:  `-2cos(sqrtx)+2/3cos^3(sqrtx)+C`