Step by step
The question
Question: solve y′ = 8x³.
y′ = 8x3
1 The idea
y′ is only about x.
So y is a function whose derivative is 8x³: an antiderivative.
y′ = 8x3: y is a function whose derivative is 8x3
2 Integrate both sides
Integrate both sides.
The left side, the integral of y′, is just y.
y = ∫ 8x3 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 = 8 × x44 + C
Tidy the numbers: y = 2x⁴ + C.
y = 2x4 + C
Careful!
Don't forget + C.
Every number C gives another solution.
Careful: not y = 2x4 ✗
The answer: y = 2x⁴ + C.
C can be any number: one solution for each C.
y = 2x4 + C
3 Check
Let's check: the derivative of our answer (the C goes, its derivative is 0) is 8x³.
The right side again.
Right!
y′ = 8x3
Let's look back.
First: the idea.
Then: integrate both sides.
Last: check.
Answer
y = 2x4 + C