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Decide on what substitution to use, and then evaluate the given integral using a substitution....

Question:

Decide on what substitution to use, and then evaluate the given integral using a substitution.

{eq}\int \frac{x}{(2x^2 - 1)^{0.4}}\ dx {/eq}

Integrals with Substitution

One technique that allows us to evaluate integrals is through substitution. This technique reverses the Chain Rule for differentiation, as it allows us to integrate composite functions. We need to be able to substitute an expression in for a variable, usually u, and the derivative of that substitution, which in that case would be du.

Answer and Explanation:

In order to use substitution to evaluate an integral, we need to find a way to turn this integral into one we can calculate. As we have an expression containing x taken to the 0.4th power in the denominator, this seems like a place to start. Let's perform the substitution as follows.

{eq}u = 2x^2 - 1\\ du = 4x dx {/eq}


The only problem with this substitution is that our original integral is missing the coefficient in the substitution for du. However, we can introduce this coefficient as long as we cancel it outside the integral by multiplying the integral by its reciprocal. This allows us to conduct this substitution, and the integral itself can be evaluated by reversing the power rule.

{eq}\begin{align*} \int \frac{x}{(2x^2 - 1)^{0.4}}\ dx &= \frac{1}{4} \frac{4x}{(2x^2 - 1)^{0.4}}\ dx\\ &= \frac{1}{4} \int \frac{1}{u^{0.4}} du\\ &= \frac{1}{4} \int u^{-0.4} du\\ &= \frac{1}{4} \cdot \frac{1}{0.6} u^{0.6} + c\\ &= \frac{5}{12}u^{0.6} + c\\ &= \frac{5}{12}(2x^2-1)^{0.6} + c \end{align*} {/eq}


Learn more about this topic:

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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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