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Chap7_Sec2

# Chap7_Sec2 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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7 TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION

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7.2 Trigonometric Integrals TECHNIQUES OF INTEGRATION In this section, we will learn: How to use trigonometric identities to integrate certain combinations of trigonometric functions.

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SINE & COSINE INTEGRALS Evaluate ∫ cos 3 x dx Simply substituting u = cos x isn’t helpful, since then du = -sin x dx . In order to integrate powers of cosine, we would need an extra sin x factor. Similarly, a power of sine would require an extra cos x factor. Example 1
Thus, here we can separate one cosine factor and convert the remaining cos 2 x factor to an expression involving sine using the identity sin 2 x + cos 2 x = 1: cos 3 x = cos 2 x . cos x = (1 - sin 2 x ) cos x Example 1 SINE & COSINE INTEGRALS

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We can then evaluate the integral by substituting u = sin x. So , du = cos x dx and 3 2 2 2 3 1 3 3 1 3 cos cos cos (1 sin )cos (1 ) sin sin x dx x x dx x x dx u du u u C x x C = × = - = - = - + = - + Example 1 SINE & COSINE INTEGRALS
SINE & COSINE INTEGRALS In general, we try to write an integrand involving powers of sine and cosine in a form where we have only one sine factor. The remainder of the expression can be in terms of cosine.

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We could also try only one cosine factor. The remainder of the expression can be in terms of sine. SINE & COSINE INTEGRALS
SINE & COSINE INTEGRALS The identity sin 2 x + cos 2 x = 1 enables us to convert back and forth between even powers of sine and cosine.

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SINE & COSINE INTEGRALS Find ∫ sin 5 x cos 2 x dx We could convert cos 2 x to 1 – sin 2 x . However, we would be left with an expression in terms of sin x with no extra cos x factor. Example 2
SINE & COSINE INTEGRALS Instead, we separate a single sine factor and rewrite the remaining sin 4 x factor in terms of cos x . So, we have: 5 2 2 2 2 2 2 2 sin cos (sin ) cos sin (1 cos ) cos sin = = - x x x x x x x x Example 2

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SINE & COSINE INTEGRALS Substituting u = cos x , we have du = sin x dx. So, 5 2 2 2 2 2 2 2 2 2 2 3 5 7 2 4 6 3 5 7 1 2 1 3 5 7 sin cos (sin ) cos sin (1 cos ) cos sin (1 ) ( ) ( 2 ) 2 3 5 7 cos cos cos = = - = - - = - - + = - - + + ÷ = - + - + x x dx x x x dx x x x dx u u du u u u u u u du C x x x C Example 2
SINE & COSINE INTEGRALS The figure shows the graphs of the integrand sin 5 x cos 2 x in Example 2 and its indefinite integral (with C = 0).

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SINE & COSINE INTEGRALS In the preceding examples, an odd power of sine or cosine enabled us to separate a single factor and convert the remaining even power.
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Chap7_Sec2 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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