Integrate ∫ (x^2 + 1/x^2) dx Step by Step

Step by step

The question

Question: find ∫ ( x² + 1/x²) dx, the antiderivative of x² + 1/x².

∫ (x² + 1x²) dx =

1 The rule

The power rule takes 1 off the power, and the power comes down in front.

x³ → 3x² (derivative)

Backwards, we undo both: add 1 to the power, then divide by the new power.

∫ and dx just mean: find the function whose derivative this is.

∫ xⁿ dx = x^{n + 1}n + 1 + C

Why + C?

A number alone has slope 0.

So any number could have been there.

We write C for it.

+ C: any number; its derivative is 0

Careful!

Every antiderivative ends with + C.

Without it, the answer is not complete.

Careful: never forget + C

2 Write as powers

To use the rule, every x needs a power.

x at the bottom is a minus power on top; a root is the power one half.

= ∫ (x² + x⁻²) dx

3 Each term

Use it on each term.

A number in front just waits.

= x³3 + x⁻¹−1 + C

Add 1 to the power: −2 + 1 is −1.

−2 + 1 = −1

So the answer is x³ over 3 − 1/x + C.

= x³3 − 1x + C

4 Check

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

The C gives 0.

We get x² + x⁻²: the question again.

Right!

(x³3 − 1x + C)′ = x² + x⁻²

Let's look back.

First: the rule.

Then: write as powers.

Then: each term.

Last: check.

Answer