MathAnalysisCalculusIntegrals

U-substitution

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Consider the following integral

$I = \int \frac{\tan(x) \sqrt{1+\sqrt[3]{\sec(x)}}}{\cos(x) \sqrt[3]{\sec(x)}} dx$ What is $I$ equal to?

Tip: You can perform several changes of variable. Just remember to reverse them all at the end! To help you get started, consider $u^3 = \cos(x)$

Select one option from the list

I = \tan(x) + C

I = 6 \cdot \sqrt{1 + \sqrt[3]{\sec(x)}} + C

I = \frac{6}{5} \left(1 + \sqrt[3]{\sec(x)} \right)^\frac{5}{2} - 2 \left(1 + \sqrt[3]{\sec(x)} \right)^\frac{3}{2} + C

I = \sqrt[3]{\sec(x)} + C

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