Integrate ∫ x^2(x^3 + 1)^2 dx

Step by step

The question

Question: find the integral of x² times (x³ + 1)².

1 Choose u

The inside: x³ + 1.

Call it u.

Its derivative is 3x², and x² 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.

We have x² dx, not 3x² dx.

x² is a third of 3x².

So x² dx is a third of du.

3 Write it with u

x³ + 1 becomes u, and x² dx becomes a third of du.

A number in front can wait outside the integral.

Careful!

x² dx is not du.

It is only a third of du.

4 Integrate

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

2 plus 1 is 3.

Multiply the numbers: 1/3 times 1/3 is 1/9.

5 Put x back

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

Put x³ + 1 back in place of u.

So the answer is (x³ + 1)³ over 9 + 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 x²(x³ + 1)²: the question again.

Right!

Let's look back.

First: choose u.

Then: find du.

Then: write it with u.

Then: integrate.

Then: put x back.

Last: check.