Step by step
The question
Question: find the integral of 2eˣ − 1/x + 4x.
1 The rules
First, e to the x.
Its derivative is eˣ itself: its slope equals its height.
So its integral is eˣ + C too.
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
e to the x stays e to the x.
The 2 waits in front.
1 over the bottom gives ln|x|.
The power rule.
Put the terms together, and add C: the slope cannot tell us a number added on the end.
So the answer is 2eˣ − ln|x| + 2x² + 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 2eˣ − 1/x + 4x: the question again.
Right!
Let's look back.
First: the rules.
Then: each term.
Last: check.