Integrate ∫ 4/(x - 3) dx

Step by step

The question

Question: find the integral of 4/(x − 3).

1 The rules

Now 1 over x.

The power rule cannot do it: x to the −1 would become x to the 0, over 0.

Instead: the derivative of ln x is 1 over x.

So the integral of 1/x is ln|x| + C.

The bars mean the size of x.

ln only takes positive numbers, and x may be negative.

With x − 3 at the bottom, it works the same: ln of the bottom.

Careful!

The power rule fails for 1/x: the new power would be 0, and we cannot divide by 0.

Use ln.

2 Integrate

4 times 1 over the bottom gives 4 ln|x − 3|.

Add C: the slope cannot tell us a number added on the end.

So the answer is 4 ln|x − 3| + C.

3 Check

Let's check: take the derivative of our answer.

ln of the bottom gives its slope over the bottom.

The numbers cancel, and we get 4/(x − 3): the question again.

Right!

Let's look back.

First: the rules.

Then: integrate.

Last: check.