If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Integral Calculus>Unit 1

Lesson 12: Definite integrals of common functions

# Definite integral of radical function

Sal finds the definite integral of 12∛x between -1 and 8 using the reverse power rule.

## Want to join the conversation?

• My calculator says -1^(4/3) is -1, giving me a final answer of 153 :/ any reason why?
• Make sure the - sign is associated with the 1, rather than negating the entire expression (calculators aren't too smart, they do what you say, not what you mean!).
-1^(4/3) = -1 because the calculator sees that expression as -(1^(4/3))
so associate the - with the 1 like so
(-1)^(4/3) = 1
• Why he didn't do anything with the 12 when taking the antiderivative of this function. Shouldn't that evaluate to 12x when taking the antiderivative?
• 12 is NOT a constant (The expression is not 12 alone, but 12x^1/3. The 12 would be a constant if it wasn't associated with any X, as in x^1/3 +12, for instance). Therefore Sal DID do something with the 12.

Taking x^1/3 alone and find its antiderivative will make you find :
`3/4x^4/3` (try taking the derivative of `3/4x^4/3` and you'll get `x^1/3`)

But we dont want that ! We want the antiderivative of 12x^1/3
So now, put your 12 in the antiderivative you've found for x^1/3 :
`12 . 3/4 . x^4/3`
and the twelve becomes the 9 you can see in the rest of the video.
• why didnt sal take the 12 up front because it is a scalar
• I would assume it was because he knew, from experience (and prior preparation), that we was going to have to use it to deal with the 4/3 that was created from the Reverse Power Rule.
• Which exponent property makes it possible to break out 1/3 from 4/3?
(1 vote)
• Differentiating it you would go from 4/3 to 1/3 because the power rule states you subtract 1 from any exponent. Since we're taking the antiderivative you would do the opposite: add 1 to the exponent, bringing it from 1/3 to 4/3.
• Can't you pull out the 12 as a coefficient
• how would this work out if it was if there was more than just x under the radical like 12* sq-rt(4-x^2) ?
(1 vote)
`https://www.khanacademy.org/math/calculus-home/integration-techniques-calc/trigonometric-substitution-calc/v/introduction-to-trigonometric-substitution`