Step by step
The question
Question: find the integral of x² + 5 over x.
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.
Powers of x use the power rule: add 1 to the power, divide by the new power.
Careful!
The power rule fails for 1/x: the new power would be 0, and we cannot divide by 0.
Use ln.
2 Each term
The power rule.
5 times 1 over the bottom gives 5 ln|x|.
Put the terms together, and add C: the slope cannot tell us a number added on the end.
So the answer is x³ over 3 + 5 ln|x| + 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 x² + 5/x: the question again.
Right!
Let's look back.
First: the rules.
Then: each term.
Last: check.