What is ∫ 3x^2(x^3 - 2)^4 dx?

Step by step

The question

Question: find the integral of 3x² times (x³ − 2)⁴.

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Choose u

The inside: x³ − 2.

Call it u.

Its derivative is 3x², and 3x² is outside.

A good sign: u′ is there, or nearly.

2 Find du

The power rule: du/dx = 3x².

Multiply both sides by dx: du = 3x² dx.

A small change in u is 3x² times a small change in x.

3 Write it with u

x³ − 2 becomes u, and 3x² dx becomes du.

4 Integrate

Integrate: add 1 to the power, divide by the new power.

4 plus 1 is 5.

5 Put x back

The question was in x, so the answer is in x too.

Put x³ − 2 back in place of u.

So the answer is (x³ − 2)⁵ over 5 + C.

6 Check

Let's check: take the derivative of our answer with the chain rule.

The power comes down and cancels the number below.

Then times the inside's slope, 3x².

That is 3x²(x³ − 2)⁴: the question again.

Right!

7 Pick the choice

Our answer is (x³ − 2)⁵ over 5 + C.

That is choice A.

8 Why not the others

B is (x³ − 2)⁵ + C.

Wrong: did not divide by 5.

C is 4(x³ − 2)³ + C.

Wrong: took the power down: integrals raise it.

D is x³(x³ − 2)⁵ over 5 + C.

Wrong: kept the 3x²: it is the du.

Let's look back.

First: choose u.

Then: find du.

Then: write it with u.

Then: integrate.

Then: put x back.

Then: check.

Then: pick the choice.

Last: why not the others.