Question

How do I evaluate the indefinite integral `intcot^5(x)*sin^4(x)dx` ?

Answer 1

`cos^2x+1/4sin^4x+lnabssinx+"c"`

Explanation:

`cot^5xsin^4x=cos^5x/sin^5xsin^4x=cos^5x/sinx=cos^4xcotx=(1-sin^2x)^2cotx=(1-2sin^2x+sin^4x)cot=cotx-2sinxcosx+sin^3xcosx`

So

`intcot^5xsin^4xdx=intcotxdx-int2sinxcosxdx+intsin^3xcosxdx`

Now let `u=sinx` and `du=cosxdx` and `v=cosx` and `dv=-sinxdx`

`intcotxdx+int-2sinxcosxdx+intsin^3xcosxdx=lnabssinx int2vdv +intu^3du=lnabssinx+v^2+1/4u^4+"c"=cos^2x+1/4sin^4x+lnabssinx+"c"`