Integrate ∫ 6/x^3 dx Step by Step

Step by step

The question

Question: find ∫ 6/x³ dx, the antiderivative of 6/x³.

∫ 6x³ 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.

= ∫ 6x⁻³ dx

3 Each term

Use it on each term.

A number in front just waits.

= 6 × x⁻²−2 + C

Add 1 to the power: −3 + 1 is −2.

−3 + 1 = −2

So the answer is −3/x² + C.

= −3x² + C

4 Check

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

The C gives 0.

We get 6 x⁻³: the question again.

Right!

(−3x² + C)′ = 6x⁻³

Let's look back.

First: the rule.

Then: write as powers.

Then: each term.

Last: check.

Answer