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Reduction Formulas for Powers of Trigonometric Functions $trig^n(x)$

Index

This article summarizes reduction formulas for the indefinite integrals of powers of trigonometric functions. All formulas are valid for integer powers $n \geq 0$ (or $n \geq 1$ for the reduction formulas), with $I_0$ and $I_1$ as base cases. Some examples using these formulas can be found here.

sin

\[\int \sin^n(x) \,\mathrm{d}x = -\dfrac{1}{n}\sin^{n-1}(x)\cos(x) + \dfrac{n-1}{n} \int \sin^{n-2}(x) \,\mathrm{d}x, \qquad n > 1\]

Let $I_n = \int \sin^n(x) \,\mathrm{d}x$, then

\[I_n = -\dfrac{1}{n}\sin^{n-1}(x)\cos(x) + \dfrac{n-1}{n} I_{n-2}, \qquad n > 1\]

Base cases:

\[\begin{aligned} I_0 &= x + C \\ I_1 &= \int \sin(x) \,\mathrm{d}x = -\cos(x) + C \end{aligned}\]

Ref: Indefinite Integrals of $\sin^n(x)$ and $\cos^n(x)$

cos

\[\int \cos^n(x) \,\mathrm{d}x = \dfrac{1}{n}\cos^{n-1}(x)\sin(x) + \dfrac{n-1}{n} \int \cos^{n-2}(x) \,\mathrm{d}x, \qquad n > 1\]

Let $I_n = \int \cos^n(x) \,\mathrm{d}x$, then

\[I_n = \dfrac{1}{n}\cos^{n-1}(x)\sin(x) + \dfrac{n-1}{n} I_{n-2}, \qquad n > 1\]

Base cases:

\[\begin{aligned} I_0 &= x + C \\ I_1 &= \int \cos(x) \,\mathrm{d}x = \sin(x) + C \end{aligned}\]

Ref: Indefinite Integrals of $\sin^n(x)$ and $\cos^n(x)$

tan

\[\int \tan^n(x) \mathrm{d}x = \dfrac{\tan^{n-1}(x)}{n-1} - \int \tan^{n-2}(x) \,\mathrm{d}x, \qquad n > 1\]

Let $I_n = \int \tan^n(x) \,\mathrm{d}x$, then

\[I_n = \dfrac{\tan^{n-1}(x)}{n-1} - I_{n-2}, \qquad n > 1\]

Base cases:

\[\begin{aligned} I_0 &= x + C \\ I_1 &= \int \tan(x) \,\mathrm{d}x = -\ln(\vert \cos(x) \vert) + C = \ln(\vert \sec(x) \vert) + C \end{aligned}\]

Ref: Indefinite Integrals of $\tan^n(x)$ and $\cot^n(x)$

cot

\[\int \cot^n(x) \mathrm{d}x = -\dfrac{\cot^{n-1}(x)}{n-1} - \int \cot^{n-2}(x) \,\mathrm{d}x, \qquad n > 1\]

Let $I_n = \int \cot^n(x) \,\mathrm{d}x$, then

\[I_n = -\dfrac{\cot^{n-1}(x)}{n-1} - I_{n-2}, \qquad n > 1\]

Base cases:

\[\begin{aligned} I_0 &= x + C \\ I_1 &= \int \cot(x) \,\mathrm{d}x = \ln(\vert \sin(x) \vert) + C = -\ln(\vert \csc(x) \vert) + C \end{aligned}\]

Ref: Indefinite Integrals of $\tan^n(x)$ and $\cot^n(x)$

sec

\[\int \sec^n(x) \mathrm{d}x = \dfrac{\sec^{n-2}(x)\tan(x)}{n-1} + \dfrac{n-2}{n-1}\int \sec^{n-2}(x)\,\mathrm{d}x, \qquad n > 1\]

Let $I_n = \int \sec^n(x) \,\mathrm{d}x$, then

\[I_n = \dfrac{\sec^{n-2}(x)\tan(x)}{n-1} + \dfrac{n-2}{n-1} I_{n-2}, \qquad n > 1\]

Base cases:

\[\begin{aligned} I_0 &= x + C \\ I_1 &= \int \sec(x) \,\mathrm{d}x = \ln(\vert \sec(x)+\tan(x) \vert) + C = - \ln(\vert \sec(x)-\tan(x) \vert) + C \end{aligned}\]

Ref: Indefinite Integrals of $\sec^n(x)$ and $\csc^n(x)$

csc

\[\int \csc^n(x) \mathrm{d}x = -\dfrac{\csc^{n-2}(x)\cot(x)}{n-1} + \dfrac{n-2}{n-1}\int \csc^{n-2}(x)\,\mathrm{d}x, \qquad n > 1\]

Let $I_n = \int \csc^n(x) \,\mathrm{d}x$, then

\[I_n = -\dfrac{\csc^{n-2}(x)\cot(x)}{n-1} + \dfrac{n-2}{n-1} I_{n-2}, \qquad n > 1\]

Base cases:

\[\begin{aligned} I_0 &= x + C \\ I_1 &= \int \csc(x) \,\mathrm{d}x = -\ln(\vert \csc(x)+\cot(x) \vert) + C = \ln(\vert \csc(x)-\cot(x) \vert) + C \end{aligned}\]

Ref: Indefinite Integrals of $\sec^n(x)$ and $\csc^n(x)$

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