Step by step
The question
Question: find the integral of 3/(x + 2) − 4e²ˣ.
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.
With e²ˣ: its derivative is 2 times e²ˣ, by the chain rule.
That 2 must not be there.
So we divide by 2.
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 + 2 at the bottom, it works the same: ln of the bottom.
Careful!
The derivative of e²ˣ is 2 times e²ˣ.
So e²ˣ alone is not the answer: divide by 2.
2 Each term
3 times 1 over the bottom gives 3 ln|x + 2|.
Divide by 2: −4/2 is −2.
Put the terms together, and add C: the slope cannot tell us a number added on the end.
So the answer is 3 ln|x + 2| − 2e²ˣ + C.
3 Check
Let's check: take the derivative of our answer.
e to the kx gives k times e to the kx.
ln of the bottom gives its slope over the bottom.
The numbers cancel, and we get 3/(x + 2) − 4e²ˣ: the question again.
Right!
Let's look back.
First: the rules.
Then: each term.
Last: check.