Evaluate the integral. Integral from 5 to 38 of 1/(fifth root of (x - 6)) dx.

Question:

Evaluate the integral.

{eq}\int_{5}^{38} \frac{1}{\sqrt[5]{x - 6}} \, \mathrm{d}x {/eq}

Integration by Substitution

Several expressions can be easily solved by substituting another variable into the equation. This variable should be related to the current one and should cancel out unnecessary variables in the integrand.

Answer and Explanation:

The expression {eq}\int_{5}^{38} \frac{1}{\sqrt[5]{x - 6}} \,dx {/eq} can be simplifies by letting {eq}u = x -6 \rightarrow du = dx {/eq}. Consequently, the bounds of integration becomes {eq}u = -1 {/eq} to {eq}u = 32 {/eq}. The expression then becomes

{eq}\begin{align*} \int_{-1}^{32} u^{-1/5} du &= \left [ \frac{5}{4} u^{4/5} \right ]_{-1}^{32} \\ &= \frac{5}{4} \left [ 32^{4/5} - (-1)^{4/5} \right ] \\ &= \frac{5}{4} \left [ 16 - 1 \right ] \\ &= \frac{75}{4} \end{align*} {/eq}

And this solves the problem.


Learn more about this topic:

Integration Problems in Calculus: Solutions & Examples

from AP Calculus AB & BC: Homework Help Resource

Chapter 13 / Lesson 13
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