Step by step
The question
Question: find the integral of 8e⁻²ˣ + 3/x − 6x².
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 e⁻²ˣ: its derivative is −2 times e⁻²ˣ, 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.
Powers of x use the power rule: add 1 to the power, divide by the new power.
Careful!
The derivative of e⁻²ˣ is −2 times e⁻²ˣ.
So e⁻²ˣ alone is not the answer: divide by −2.
2 Each term
Divide by −2: 8 over −2 is −4.
3 times 1 over the bottom gives 3 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 −4e⁻²ˣ + 3 ln|x| − 2x³ + 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 8e⁻²ˣ + 3/x − 6x²: the question again.
Right!
Let's look back.
First: the rules.
Then: each term.
Last: check.