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`int cos^3xsin^4x dx` Find the indefinite integral

`intcos^3(x)sin^4(x)dx`


Rewrite the integral as ,


`intcos^3(x)sin^4(x)dx=intcos^2(x)cos(x)sin^4(x)dx`


Now use the trigonometric identity:`cos^2(x)=1-sin^2(x)`


`=int(1-sin^2(x))cos(x)sin^4(x)dx`


Apply integral substitution: `u=sin(x)`


`du=cos(x)dx`


`=int(1-u^2)u^4du`


`=int(u^4-u^6)du`


Apply the sum and power rules,


`=intu^4du-intu^6du`


`=(u^(4+1)/(4+1))-(u^(6+1)/(6+1))`


`=u^5/5-u^7/7`


Substitute back u=sin(x)  and add a constant C to the solution,


`=(sin^5(x))/5-(sin^7(x))/7+C`

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