Step by step
The question
Question: solve dy/dx = 3x² − 2.
dydx = 3x2 − 2
1 The idea
dy/dx is only about x.
So y is a function whose derivative is 3x² − 2: an antiderivative.
y′ = 3x2 − 2: y is a function whose derivative is 3x2 − 2
2 Integrate both sides
Integrate both sides.
The left side, the integral of dy/dx, is just y.
y = ∫ (3x2 − 2) dx
The power rule backwards: add 1 to the power, divide by the new power.
∫ xn dx = xn + 1n + 1
Each term by its rule, and one + C: the number we cannot see after a derivative.
y = 3 × x33 − 2 × x1 + C
Tidy the numbers: y = x³ − 2x + C.
y = x3 − 2x + C
Careful!
Don't forget + C.
Every number C gives another solution.
Careful: not y = x3 − 2x ✗
The answer: y = x³ − 2x + C.
C can be any number: one solution for each C.
y = x3 − 2x + 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!
y′ = 3x2 − 2
Let's look back.
First: the idea.
Then: integrate both sides.
Last: check.
Answer
y = x3 − 2x + C