Question

What is the antiderivative of `x/(x^2+1)`?

Answer 1

`1/2ln(x^2+1)+C, or, ln (x^2+1)^(1/2)+C, or, lnsqrt(x^2+1)+C`

Explanation:

Let `I=intx/(x^2+1)dx`.

Subst. `(x^2+1)=t rArr 2xdx=dt rArr xdx=dt/2`.

`:. I=int1/t*dt/2=1/2int1/tdt=1/2ln|t|`

`:. I=1/2ln(x^2+1)+C, or, ln (x^2+1)^(1/2)+C, or, lnsqrt(x^2+1)+C`.