KindiakMath

The Classic Secant Integral

No course in calculus is complete without the famous problem of integrating \sec^3.

Problem 1. Evaluate \displaystyle \int \sec^3(x)\, \mathrm dx.

(Click for Solution)

Solution. Write \sec^3(x) = \sec^2(x) \sec(x). Since \sec^2(x) integrates to \tan(x), we integrate by parts to obtain

\displaystyle \begin{aligned} \int \sec^3(x)\, \mathrm dx &= \int \sec^2(x) \sec(x)\, \mathrm dx \\ &= \underbrace{\tan(x)}_{\text I} \underbrace{\sec(x)}_{\text S} - \int \underbrace{\tan(x)}_{\text I} \cdot \underbrace{\sec(x)\tan(x)}_{\text D}\,\mathrm dx. \end{aligned}

Writing \tan^2(x) = \sec^2(x) - 1,

\displaystyle \begin{aligned} \int \sec^3(x)\, \mathrm dx &= \tan(x) \sec(x) - \int \tan(x) \cdot \sec(x)\tan(x)\,\mathrm dx \\ &= \tan(x) \sec(x) - \int (\sec^2(x) - 1) \cdot \sec(x)\,\mathrm dx \\ &= \tan(x) \sec(x) - \int \sec^3(x)\, \mathrm dx + \int \sec(x)\,\mathrm dx. \end{aligned}

By algebra and recalling our result for the integral of \sec,

\displaystyle \begin{aligned} \int \sec^3(x)\, \mathrm dx &= \frac 12 \tan(x) \sec(x) + \frac 12 \int \sec(x)\,\mathrm dx \\ &= \frac 12 \tan(x) \sec(x) + \frac 12 \ln|{\sec(x) + \tan(x)}| + C. \end{aligned}

—Joel Kindiak, 28 Oct 24, 1042H

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