Question

How do you use Integration by Substitution to find `intx/(x^2+1)dx`?

Answer 1

`intx/(x^2+1)dx=1/2ln(x^2+1)+"c"`

Explanation:

We want of find `intx/(x^2+1)dx`.

We make the natural substitution `u=x^2+1` and `du=2xdx`.

So

`intx/(x^2+1)dx=1/2int1/udu=1/2lnabsu= 1/2lnabs(x^2+1)+"c"`

We notice that `x^2+1>0` so we don't need to take the absolute value. So the final answer is

`1/2ln(x^2+1)+"c"`

Answer 2

You don't. You simply write down the answer.

Explanation:

`int x/(x^2+1)\ dx`
`=(1/2)int (2x)/(x^2+1)`
The integral is of the form `int (f'(x))/f(x)\ dx=ln|f(x)|`
So you can just write down the answer, dropping the `||`, as `x^2+1>0`:

`1/2ln(x^2+1)+c`
or alternatively `lnsqrt(x^2+1)+c`
With practice you can do this type in one step at sight!