Step by step
The question
Question: find ∫ (8 x³ − 6x + 5) dx, the antiderivative of 8 x³ − 6x + 5.
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.
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.
Why + C?
A number alone has slope 0.
So any number could have been there.
We write C for it.
Careful!
Every antiderivative ends with + C.
Without it, the answer is not complete.
2 Each term
Use it on each term.
A number alone, like 5, comes from 5x: it gets an x.
Divide the numbers where we can: 8 divided by 4 is 2, 6 divided by 2 is 3.
So the answer is 2 x⁴ − 3 x² + 5x + C.
3 Check
Let's check: take the derivative of our answer.
The C gives 0.
We get 8 x³ − 6x + 5: the question again.
Right!
4 Pick the choice
Our answer is 2x⁴ − 3x² + 5x + C.
That is choice B.
5 Why not the others
A is 24x² − 6 + C.
Wrong: differentiated instead.
C is 8x⁴ − 6x² + 5x + C.
Wrong: did not divide by the new powers.
D is 2x⁴ − 3x² + C.
Wrong: lost the 5x: 5 gives 5x.
Let's look back.
First: the rule.
Then: each term.
Then: check.
Then: pick the choice.
Last: why not the others.