Integration using trigonometric identities

Integration using trigonometric identities

In this section, we will take a look at several methods for integrating trigonometric functions. All methods require us to use U-Substitution and substituting with trigonometric identities. In addition, the trigonometric functions we are dealing with are products of sine and cosine with powers. There are a total of 3 cases. The first case is when the power of cosine is odd. The second case is when the power of sine is odd. Lastly, the third case is when both the powers of sine and cosine are even.
Basic Concepts: Other word problems relating angles in trigonometry, Double-angle identities, Pythagorean identities, Quotient identities and reciprocal identities, Sum and difference identities, Cofunction identities

Lessons

Pre-requisite: * Trigonometry –"Trigonometric Identities"

Note: Strategy for evaluating ∫sin⁡■x\int \sin^{\blacksquare}x cos⁡■x\cos^{\blacksquare}x dxdx

Case 1: ∫sin⁡■x\int \sin^{\blacksquare}x cos⁡oddx\cos^{odd}x dxdx
1. strip out one cosinecosine factor
2. express the remaining cosinecosine factors in terms of sinesine using the Pythagorean Identity: cos⁡2x=1−sin⁡2x\cos^2x=1-\sin^2x
3. substitute u=sin⁡xu=\sin x

Case 2: ∫sin⁡oddx\int \sin^{odd}x cos⁡■x\cos^{\blacksquare}x dxdx
1. strip out one sinesine factor
2. express the remaining sinesine factors in terms of cosinecosine using the Pythagorean Identity: sin⁡2x=1−cos⁡2x\sin^2x=1-\cos^2x
3. substitute u=cos⁡xu=\cos x

Case 3: ∫sin⁡evenx\int \sin^{even}x cos⁡evenx\cos^{even}x dxdx
1. use the half-angle identities: sin⁡2x=12(1−cos⁡2x)\sin^2x=\frac{1}{2}(1-\cos2x) oror cos⁡2x=12(1+cos⁡2x)\cos^2x=\frac{1}{2}(1+\cos2x)
2. if necessary, use the double-angle identity: sin⁡x\sin x cos⁡x\cos x =12sin⁡2x=\frac{1}{2}\sin2x

  • 1.
    Evaluate the integral (odd power of cosine).
    a)
    ∫sin⁡8x\int \sin^8x cos⁡3x\cos^3 x dxdx

  • 2.
    Evaluate the integral (odd power of sine).
    a)
    ∫sin⁡5x\int \sin^5x cos⁡2x\cos^2 x dxdx

  • 3.
    Evaluate the integral (even powers of sine/cosine).
    a)
    ∫sin⁡2x\int \sin^2x dxdx

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6.
Integration Techniques
6.1
U-Substitution
6.2
Integration using trigonometric identities
6.3
Trigonometric substitution
6.4
Integration by parts
6.5
Integration of rational functions by partial fractions
6.6
Numerical Integration
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