Step By Step Calculus » 15.4 - Integration for Trigonometric Functions

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Synopsis
Change of Variables, Trigonometric Identities, and Integration by Parts help in integrating trigonometric functions. One can use the following guidelines for trigonometric functions integrals. For problems of the following type

\displaystyle {\textbf{(Problem I)}\quad \int \sin^m x\cos^n xdx\quad\textrm{with at least one of $m$ or $n$ being a positive integer,} }\displaystyle {\textbf{(Problem I)}\quad \int \sin^m x\cos^n xdx\quad\textrm{with at least one of $m$ or $n$ being a positive integer,} }

we use the following procedure.
Step 1: (Convert to Odd Power) If both mm and nn are positive even integers, then we use the following formulae to convert the given function into a function of only \sin\sin or only \cos\cos with odd power.
\displaystyle \sin^2 x =\frac{1}{2}(1-\cos 2x) \qquad \cos^2x=\frac{1}{2}(1+\cos 2x) \qquad \sin x\cos x =\frac{1}{2}\sin 2x
\displaystyle \sin^2 x =\frac{1}{2}(1-\cos 2x) \qquad \cos^2x=\frac{1}{2}(1+\cos 2x) \qquad \sin x\cos x =\frac{1}{2}\sin 2x
Then, apply the Linearity Property of integration to treat integration of each parts differently.
Step 2: (Apply \mathbf{\sin^2\theta+\cos^2\theta=1}\mathbf{\sin^2\theta+\cos^2\theta=1}) If \sin\sin (or \cos\cos) has odd power, we keep a single factor of \sin\sin (or \cos\cos) separate and transform the remaining factors into \cos\cos (or \sin\sin) respectively using the trigonometric identity (\sin^2 \theta+\cos^2\theta=1\sin^2 \theta+\cos^2\theta=1) so that the separated single factor can play the role of g^{\prime}(x)g^{\prime}(x) in the Change of Variables.
Step 3: (Change Variable) If we have separated a single factor of \sin\sin (or \cos\cos) in Step 2, then substitute a variable tt for \cos\cos (or \sin\sin).
Step 4: (Integrate) Use the Linearity Property, Change of Variables, and the simple formulae for integration as needed to obtain the final result.
For problems of the following type

\displaystyle { \textbf{(Problem II)}\quad \int \tan^mx\sec^nxdx \;\textrm{with at least one of $m$ or $n$ being a positive integer,} }\displaystyle { \textbf{(Problem II)}\quad \int \tan^mx\sec^nxdx \;\textrm{with at least one of $m$ or $n$ being a positive integer,} }

we use the following procedure.
Step 1: (Convert \tan^2\to\sec^2\tan^2\to\sec^2) We express all factors of \tan^2 x\tan^2 x in terms of \sec^2 x\sec^2 x using the identity \tan^2x=\sec^2x-1\tan^2x=\sec^2x-1 when either (Case i) \tan\tan has a positive odd power and \sec\sec has a nonzero power, or (Case ii) \tan\tan has an integer power \geq 1\geq 1 and \sec\sec has a zero power, or (Case iii) \tan\tan has a positive even power and \sec\sec has an integer power. Examples of applicable cases are:
\displaystyle \displaystyle (Case i) \displaystyle \int\tan^3x\sec^{-3/2}x dx,\quad \displaystyle \int\tan^3x\sec^{-3/2}x dx,\quad (Case ii) \displaystyle \int \tan^4 x dx, \quad\displaystyle \int \tan^4 x dx, \quad(Case iii) \displaystyle \int \tan^2 x \sec^3 x dx \displaystyle \int \tan^2 x \sec^3 x dx
Then, apply the Linearity Property of integration to treat integration of each parts differently.
Step 2: (Convert \sec^2\to\tan^2\sec^2\to\tan^2) If we separate out a single factor of \sec x \tan x\sec x \tan x, we discover that the remaining term is already expressed in terms of \sec x\sec x. Then we can proceed to Step 3. Otherwise, if \sec\sec has a positive even power, then separate out a single factor of \sec^2 x\sec^2 x and express remaining factors of \sec^2 x\sec^2 x in terms of \tan^2 x\tan^2 x using the identity \sec^2x=1+\tan^2x\sec^2x=1+\tan^2x.
Step 3: (Change Variable) With the separated \sec x \tan x\sec x \tan x or \sec^2x\sec^2x, we can substitute a variable tt for \sec x\sec x or \tan x\tan x respectively.
Step 4: (Integrate) Use the Linearity Property, Change of Variables, and the simple formulae for integration as needed. In the case \sec\sec has a positive odd power and \tan\tan has zero power (i.e. \tan\tan does not exist), use the reduction formula obtained from integration by parts.
For problems of the following type

\displaystyle { \textbf{(Problem III)} \quad \int \cot^mx\csc^nxdx \;\textrm{with at least one of $m$ or $n$ being a positive integer,} }\displaystyle { \textbf{(Problem III)} \quad \int \cot^mx\csc^nxdx \;\textrm{with at least one of $m$ or $n$ being a positive integer,} }

we use the following procedure
Step 1: (Convert \cot^2\to\csc^2\cot^2\to\csc^2) We express all factors of \cot^2 x\cot^2 x in terms of \csc^2 x\csc^2 x using the identity \cot^2x=\csc^2x-1\cot^2x=\csc^2x-1 when either (Case i) \cot\cot has a positive odd power and \csc\csc has a nonzero power, or (Case ii) \cot\cot has an integer power \geq 1\geq 1 and \csc\csc has a zero power, or (Case iii) \cot\cot has a positive even power and \csc\csc has an integer power. Examples of applicable cases are:
\displaystyle \displaystyle (Case i) \displaystyle \int\cot^3x\csc^{1/2}x dx,\quad \displaystyle \int\cot^3x\csc^{1/2}x dx,\quad (Case ii) \displaystyle \int \cot^3 x dx, \quad\displaystyle \int \cot^3 x dx, \quad(Case iii) \displaystyle \int \cot^2 x \csc^4 x dx \displaystyle \int \cot^2 x \csc^4 x dx
Then, apply the Linearity Property of integration to treat integration of each parts differently.
Step 2: (Convert \csc^2\to\cot^2\csc^2\to\cot^2) If we separate out a single factor of \csc x \cot x\csc x \cot x, we find that the remaining term is already expressed in terms of \csc x\csc x. Then we go to Step 3. Otherwise, if \csc\csc has a positive even power, then we separate out a single factor of \csc^2 x\csc^2 x and express remaining factors of \csc^2 x\csc^2 x in terms of \cot^2 x\cot^2 x using the identity \csc^2x=1+\cot^2x\csc^2x=1+\cot^2x.
Step 3: (Change Variable) With the separated \csc x \cot x\csc x \cot x or \csc^2x\csc^2x, we csn substitute a variable tt for \csc x\csc x or \cot x\cot x respectively.
Step 4: (Integrate) Use the Linearity Property, Change of Variables, and the simple formulae for integration as needed. In the case \csc\csc has a positive odd power and \cot\cot has zero power (i.e. \cot\cot does not exist), use the reduction formula obtained from integration by parts.
For problems of the following type

\displaystyle { \textbf{(Problem IV)}\quad \int \sin mx\cos nx dx, \;\textrm{or}\; \int \sin mx\sin nx dx, \;\textrm{or}\; \int \cos mx\cos nx dx, }\displaystyle { \textbf{(Problem IV)}\quad \int \sin mx\cos nx dx, \;\textrm{or}\; \int \sin mx\sin nx dx, \;\textrm{or}\; \int \cos mx\cos nx dx, }

we use the following procedure.
Step 1: (Convert Product to Sum) Using the following formulae, express the given product from into a summation form.
\displaystyle \sin mx \cos nx\displaystyle \sin mx \cos nx
\displaystyle =\displaystyle =
\displaystyle \frac{1}{2}\left[ \sin (m-n)x+\sin (m+n)x\right] \displaystyle \frac{1}{2}\left[ \sin (m-n)x+\sin (m+n)x\right]
\displaystyle \sin mx \sin nx\displaystyle \sin mx \sin nx
\displaystyle =\displaystyle =
\displaystyle \frac{1}{2}\left[ \cos (m-n)x-\cos (m+n)x\right] \displaystyle \frac{1}{2}\left[ \cos (m-n)x-\cos (m+n)x\right]
\displaystyle \cos mx \cos nx\displaystyle \cos mx \cos nx
\displaystyle =\displaystyle =
\displaystyle \frac{1}{2}\left[ \cos (m-n)x+\cos (m+n)x\right] \displaystyle \frac{1}{2}\left[ \cos (m-n)x+\cos (m+n)x\right]
Step 2: (Integrate) Use the Linearity Property, Change of Variables, and the simple formulae for integration as needed.