Step by step
The question
Question: find ∫ √x dx, the antiderivative of √x.
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 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.
3 Each term
Use it on each term.
A number in front just waits.
Add 1 to the power: 1/2 + 1 is 3/2.
So the answer is 2/3 × x^(3/2) + C.
4 Check
Let's check: take the derivative of our answer.
The C gives 0.
We get x^(1/2): the question again.
Right!
Let's look back.
First: the rule.
Then: write as powers.
Then: each term.
Last: check.