2.3 Integration using trigonometric identities

Integration using trigonometric identities


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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