Which is ∫ (6x^2 - 4x + 1) dx? Step by Step

Step by step

The question

Question: find ∫ (6 x² − 4x + 1) dx, the antiderivative of 6 x² − 4x + 1.

∫ (6x² − 4x + 1) dx

There are four choices.

First we solve it, the normal way.

Then we pick.

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 Each term

Use it on each term.

A number alone, like 1, comes from 1x: it gets an x.

= 6 × x³3 − 4 × x²2 + x + C

Divide the numbers where we can: 6 divided by 3 is 2, 4 divided by 2 is 2.

6 ÷ 3 = 2

4 ÷ 2 = 2

So the answer is 2 x³ − 2 x² + x + C.

= 2x³ − 2x² + x + C

3 Check

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

The C gives 0.

We get 6 x² − 4x + 1: the question again.

Right!

(2x³ − 2x² + x + C)′ = 6x² − 4x + 1

4 Pick the choice

Our answer is 2x³ − 2x² + x + C.

That is choice D.

D) 2x³ − 2x² + x + C

5 Why not the others

A is 12x − 4 + C.

Wrong: differentiated instead.

A) 12x − 4 + C ✗ differentiated instead

B is 6x³ − 4x² + x + C.

Wrong: did not divide by the new power.

B) 6x³ − 4x² + x + C ✗ did not divide by the new power

C is 2x³ − 4x² + x + C.

Wrong: 4x gives 2x², not 4x².

C) 2x³ − 4x² + x + C ✗ 4x gives 2x^2, not 4x^2

Let's look back.

First: the rule.

Then: each term.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer