Calculus Small Changes, Big Ideas

Section 8.5 Advanced integration strategy

• How do you know what to try? Recognizing structural signals: composition (substitution), products (parts), radicals of quadratics (trig substitution), rational functions (partial fractions), and algebraic simplification before applying any method.

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• Choosing what powers to set apart in trigonometric integrals: handling odd and even powers of sine and cosine; strategies for powers of tangent and secant; rewriting with identities before integrating.

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  • For \(\int \sin^{m} x\cos^{n} x\dd{x}\text{:}\)

    • If either \(m\) or \(n\) is odd, save one factor and convert the rest using the identity \(\sin^{2} x+\cos^{2} x=1\text{.}\)

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    • If both \(m\) and \(n\) are even, use the power reduction identities

      \begin{equation*} \sin^{2} x=\tfrac12-\tfrac12\cos 2x,\qquad \cos^{2} x=\tfrac12+\tfrac12\cos 2x\text{.} \end{equation*}

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  • For \(\int \tan^{m} x\sec^{n} x\dd{x}\text{:}\)

    • If \(m\) is even, save a factor of \(\sec^{2} x\) and use \(\sec^{2} x=1+\tan x\) to rewrite the integral.

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    • If \(n\) is odd, save a factor of \(\sec x\tan x\) and use \(\tan^{2} x=\sec^{2} x-1\) to rewrite the integral.

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    • If \(m\) is odd and \(n\) is even, use substitution and integration by parts as necessary. The integral of \(\sec x\) comes up in analyzing the Mercator projection!

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• Combining multiple techniques in one problem: substitution followed by parts, long division before partial fractions, completing the square before trig substitution, and back-substitution.

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• Integration cycles and solving for the integral: applying integration by parts twice and rearranging algebraically. Example: \(\DS\int \ee^{x}\cos x\dd{x}\)

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• Reduction formulas: defining integrals like \(I_n = \int \sin^n x\dd{x}\) or \(I_n = \int x^n e^x \dd{x}\) and expressing them recursively in terms of lower powers.

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  \begin{align*} \int x^n \ee^{ax} \dd{x} \amp = \frac1a\qty(x^n \ee^{ax} - n\int x^{n-1} \ee^{ax} \dd{x}) \\ \int \cos^n x\dd{x} \amp = \frac1n\cos^{n-1}x\sin x+\frac{n-1}{n}\int\cos^{n-2}x\dd{x} \\ \int \sin^n x\dd{x} \amp = -\frac1n\sin^{n-1}x\cos x+\frac{n-1}{n}\int\sin^{n-2}x\dd{x} \\ \int \tan^n x\dd{x} \amp = \frac{1}{n-1}\tan^{n-1}x-\int\tan^{n-2}x\dd{x} \\ \int \cot^n x\dd{x} \amp = -\frac{1}{n-1}\cot^{n-1}x-\int\cot^{n-2}x\dd{x} \\ \int \sec^n x\dd{x} \amp = \frac{1}{n-1}\sec^{n-1}x\sin x+\frac{n-2}{n-1}\int\sec^{n-2}x\dd{x} \\ \int \csc^n x\dd{x} \amp = -\frac{1}{n-1}\csc^{n-1}x\cos x+\frac{n-2}{n-1}\int\csc^{n-2}x\dd{x} \end{align*}
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• Looking at an integral table

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