Integrate the following functions w.r.t.`x` ,
`(i) int(ax+b)^(n)dx` `(ii)int(2^((1)/(x)))/(x^(2))dx`
`(iii) int(2dx)/((e^(x)+e^(-x))^(2))`
`(iv) (1)/(x^(2)(x^(4)+1)^((3)/(4))dx`
`(v) int((tan^(-1)x)^(3))/(1+x^(2))dx`

Explanation

`(i)` Let, `l=int(ax+b)^(n)dx`
`{:("put"ax+b=t),(impliesadx=dt):}}impliesl=int(t^(n))/(a)dt=(1)/(a)(t^(n+1))/(n+1)+C`
`:.l=((ax+b)^(n+1))/(a(n+1))+C`
`(ii) l=int(2^((1)/(x)))/(x^(2))dx`, put `(1)/(x)=t`
`implies-(1)/(x^(2))dx=dt`
`:.l=-int2^(t)dt=(-2^(t))/(log2)+Cimpliesl=(-2^((1)/(x)))/(log2)+C`
`(iii) l=int(2)/((e^(x)+e^(-x))^(2))dx`
`:.l=int(2e^(2x)/((e^(2x)+1)^(2))dx`, put `e(2x)+1=t`
`2e^(2x)dx=dt`
`:.l=int(dt)/(t^(2))impliesl=-(1)/(t)+Cimpliesl=(-1)/(e^(2x)+1)+C`
`:.l=(-e^(-x))/(e^(x)+e^(-x))+C`
`(iv) l=int(1)/(x^(2)(x^(4)+1)^((3)/(4)))dx`
`impliesl=int(1)/(x^(5){1+(1)/(x^(4))}^((3)/(4)))dx`, put `1+(1)/(x^(4))=t`
`implies(-4)/(x^(5))dx=dt`
`:.l=(-1)/(4)int(dt)/(t^((3)/(4)))impliesl=(-1)/(4)(t^((1)/(4)))/((1)/(4))impliesl=-{1+(1)/(x^(4))}^((1)/(4))+C`
`(v) I=int((tan^(-1)x)^(3))/(1+x^(2))dx`, put `tan^(-1)x=t`
`implies(1)/(1+x^(2))dx=dt`
`:.l=int t^(3)dtimpliesl=(t^(4))/(4)+Cimpliesl=({tan^(-1)x}^(4))/(4)+C`