Step by step
The question
Question: find the integral of 10/(5x + 2).
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 5x + 2 at the bottom: ln of the bottom, divided by 5.
Its derivative would bring a 5 out, by the chain rule.
Careful!
The power rule fails for 1/x: the new power would be 0, and we cannot divide by 0.
Use ln.
2 Integrate
Divide by 5: 10/5 is 2.
Add C: the slope cannot tell us a number added on the end.
So the answer is 2 ln|5x + 2| + 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 10/(5x + 2): the question again.
Right!
Let's look back.
First: the rules.
Then: integrate.
Last: check.