Evaluate the interval, if possible. If an antiderivative cannot be found, state that it cannot be...

Question:

Evaluate the interval, if possible. If an antiderivative cannot be found, state that it cannot be integrated.

{eq}\int \sqrt[4]{r^3 + 1}dr {/eq}

Evaluating an Integral

The indefinite integral of a function {eq}f(x) {/eq} is denoted by {eq}\displaystyle \int f(x)dx {/eq} and represents the family of functions which are antiderivatives of {eq}f(x). {/eq} This means that {eq}\displaystyle\int f(x)dx = F(x) + c {/eq} where {eq}c {/eq} is a constant and {eq}F'(x) = f(x). {/eq} One common rule for finding an integral is the power rule, which states that {eq}\displaystyle\int x^n dx = \frac{1}{n+1}x^{n+1} + c {/eq}

Answer and Explanation:

The integral {eq}\displaystyle\int\sqrt[4]{r^3+1}dr {/eq} cannot be integrated. One might try to use substitution on this integral by setting {eq}u=r^3 + 1, {/eq} however then {eq}du=3r^2dr {/eq} and we cannot rewrite the integral so that it only contains u's and not r's. The power rule for integration does not apply here, and integration by parts and trigonometric substitution also do not work.

The antiderivative cannot be found using standard calculus techniques - a computer algebra system should be used.


Learn more about this topic:

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Antiderivative: Rules, Formula & Examples

from Calculus: Help and Review

Chapter 8 / Lesson 12
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