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.


Pre-requisite: * Trigonometry –“Trigonometric Identities”

Note: Strategy for evaluating sinx\int \sin^{\blacksquare}x cosx\cos^{\blacksquare}x dxdx

Case 1: sinx\int \sin^{\blacksquare}x cosoddx\cos^{odd}x dxdx
1. strip out one cosinecosine factor
2. express the remaining cosinecosine factors in terms of sinesine using the Pythagorean Identity: cos2x=1sin2x\cos^2x=1-\sin^2x
3. substitute u=sinxu=\sin x

Case 2: sinoddx\int \sin^{odd}x cosx\cos^{\blacksquare}x dxdx
1. strip out one sinesine factor
2. express the remaining sinesine factors in terms of cosinecosine using the Pythagorean Identity: sin2x=1cos2x\sin^2x=1-\cos^2x
3. substitute u=cosxu=\cos x

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

  • 1.
    Evaluate the integral (odd power of cosine).
    • b)
      cos5x\int \cos^5x dxdx
  • 2.
    Evaluate the integral (odd power of sine).
    • b)
      π6π2sin5x\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \sin^5x cos3x\cos^3 x dxdx
    • c)
      sin3x\int \sin^3x cosx\sqrt{\cos x} dxdx
  • 3.
    Evaluate the integral (even powers of sine/cosine).
    • b)
      cos4x\int \cos^4x dxdx
    • c)
      sin4x\int \sin^4x cos2x\cos^2 x dxdx
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Integration using trigonometric identities

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