Integrate ∫ 4/x^2 dx Step by Step

Step by step

The question

Question: find ∫ 4/x² dx, the antiderivative of 4/x².

∫ 4x² 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.

= ∫ 4x⁻² dx

3 Each term

Use it on each term.

A number in front just waits.

= 4 × x⁻¹−1 + C

Add 1 to the power: −2 + 1 is −1.

−2 + 1 = −1

So the answer is −4/x + C.

= −4x + C

4 Check

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

The C gives 0.

We get 4 x⁻²: the question again.

Right!

(−4x + C)′ = 4x⁻²

Let's look back.

First: the rule.

Then: write as powers.

Then: each term.

Last: check.

Answer