Question

How do I evaluate `int(3x)/(x^2+1)dx`?

Answer 1

Use substitution (often called `u` substitution).

Notice that the derivative of the denominator is `2x`, which is a lot like what's in the numerator. The `3` in the numerator is kinda in our way, so move it outside the integral sign.

(Recall that `intcf(x)dx=cintf(x)dx`, so this wont't change the integral.)

Now the integral is `3intx/(x^2+1)dx`.

Let `u=x^2+1` making `du=2xdx`and proceed with whatever details of substitution you've learned.

I use: `3intx/(x^2+1)dx=3int1/2(2x)/(x^2+1)dx`

`=3/2int1/(x^2+1)2xdx=3/2int1/udu`

You can probably see how to finish to get `3/2ln(x^2+1)+C`.