Step by step
The question
Question: solve dy/dx = 3x² − 2.
When y′ is only about x, we integrate.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The idea
dy/dx is only about x.
So y is a function whose derivative is 3x² − 2: an antiderivative.
2 Integrate both sides
To undo the derivative, we integrate both sides.
The left side, the integral of dy/dx, is just y.
The power rule backwards: add 1 to the power, divide by the new power.
Each term by its rule, and one + C: the number we cannot see after a derivative.
Tidy the numbers: y = x³ − 2x + C.
Careful!
Don't forget + C.
Every number C gives another solution.
The right way: y = x³ − 2x + C.
The answer: y = x³ − 2x + C.
C can be any number: one solution for each C.
3 Check
Let's check: the derivative of our answer (the C goes, its derivative is 0) is 3x² − 2.
The right side again.
Right!
4 Pick the choice
Our answer is y = x³ − 2x + C.
That is choice C.
5 Why not the others
A is 6x + C.
Wrong: differentiated instead.
B is x³ − 2 + C.
Wrong: −2 gives −2x.
D is 3x³ − 2x + C.
Wrong: did not divide by the new power.
Let's look back.
First: the idea.
Then: integrate both sides.
Then: check.
Then: pick the choice.
Last: why not the others.