Secant Substitution: x cubed times radical: Find .

Here we can use the Secant Substitution Rule with :

Let , ,

and using the Tangent-Secant identity, .

Upon substitution, we have

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Our strategy now is to convert to an integral of powers of the tangent, saving 2 factors of secant for the differential of the tangent. We use the Tangent-Secant identity again:

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Regrouping, we get

Now we can substitute , , to get

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Using the Power Law for Integrals, we get

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We can use our definition of u above to go back to an expression in terms of the tangent:

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Using the Secant Diagram, we can convert back to an expression in x:

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