132Lecture7.pdf - Lecture 7 Trigonometric Integration 1...

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Lecture 7: Trigonometric Integration 1 Some indentities In this section we deal with the integration of trigonometric functions. First we will review some identities: 1. sin 2 ( x ) + cos 2 ( x ) = 1 . 2. 1 + tan 2 ( x ) = sec 2 ( x ) . 3. Half-angle formula: sin( x ) cos( x ) = sin(2 x ) 2 . 4. Half-angle formula: cos 2 ( x ) = 1+cos(2 x ) 2 . 5. Half-angle formula: sin 2 ( x ) = 1 - cos(2 x ) 2 . 6. sin( A ) cos( B ) = sin( A - B )+sin( A + B ) 2 . 7. sin( A ) sin( B ) = cos( A - B ) - cos( A + B ) 2 . 8. cos( A ) cos( B ) = cos( A - B )+cos( A + B ) 2 . The following integrals are quite useful: 1. R cos( kx ) dx = sin( kx ) k + C. 2. R sin( kx ) dx = - cos( kx ) k + C. 2 R sin m ( x ) cos n ( x ) dx We look at some integrals involving powers of sin( x ) and cos( x ). Depending on the powers, they have different techniques. 1. Power of sin( x ) is odd. (a) Factor out one power of sin( x ) ; (b) Replacing the remaining even power of sin( x ) with sin 2 ( x ) = 1 - cos 2 ( x ) ; (c) Make a u-substitution u = cos( x ) and integrate. Evaluate the integral R sin 3 ( x ) dx . As the power of sin( x ) is odd, saving one sin( x ) from sin 3 ( x ), therefore Z sin 3 ( x ) dx = Z sin 2 ( x ) sin( x ) dx. 1
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Make a substitution by u = cos( x ) and du = - sin( x ) dx , therefore Z sin 3 ( x ) dx = Z sin 2 ( x ) sin( x ) dx = Z (1 - cos 2 ( x )) sin( x ) dx = - Z (1 - u 2 ) du = - u - u 3 3 + C = - cos( x ) - cos 3 ( x ) 3 + C.
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  • Summer '08
  • WOLBACH
  • Calculus, Berlin U-Bahn, dx

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