Question

How do you find the antiderivative of `(5x^2)/(x^2 + 1)`?

Answer 1

I would start by factorizing out `x^2` from the denominator:

`int(5x^2)/(x^2+1)dx=int(5x^2)/(x^2(1+1/x^2))dx=int5/(1+1/x^2)dx=`

now set: `1/x^2=t^2`
`x=1/t` and `dx=-1/t^2dt`

The integral becomes:

`5int1/(1+t^2)*(-1/t^2)dt=-5[int1/(t^2(1+t^2))dt]=`

`=-5{int[1/t^2-1/(1+t^2)]dt}=`

`=-5[t^(-1)/-1-arctan(t)]=` going back to `x`

`5x+5arctan(1/x)+c`