Question

How do you find the antiderivative of `int 1/(x^2+3x) dx`?

Answer 1

`1/3ln|x|-1/3ln|x+3|+C`.

Explanation:

Write down:
`=int1/(x(x+3))dx`
`=int( { }/x + { }/(x+3))dx`.

Then apply the cover-up rule for partial fractions. To find out what goes over the `x`, use your finger to cover up the factor `x` in the denominator of the fraction on the first line, and replace all other `x`'s with zero. Similarly, to find out what goes over the `x+3`, cover up the `x+3` and replace the other `x` with `-3`. In each case, you replace `x` with whatever value of `x` makes the expression under your finger zero.

`=int( (1)/(0+3))/x + (1/(-3))/(x+3)dx`
`=1/3int1/x-1/(x+3)dx`.