Trigonometric Integrals - Bruce E. Shapiro
Trigonometric Integrals - Bruce E. Shapiro
Trigonometric Integrals - Bruce E. Shapiro
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TOPIC 7. TRIGONOMETRIC INTEGRALS Math 150A<br />
∫<br />
Example 7.3<br />
sin 5 x dx<br />
∫<br />
∫<br />
sin 5 x dx = sin x(sin 2 x) 2 dx<br />
(7.15a)<br />
∫<br />
= sin x(1 − cos 2 x) 2 dx<br />
(7.15b)<br />
= −<br />
∫<br />
(1 − u 2 ) 2 du with u = cos x (7.15c)<br />
= −<br />
∫<br />
(1 − 2u 2 + u 4 )du (7.15d)<br />
= −u + 2 3 u3 − 1 5 u5 + C (7.15e)<br />
= − cos x + 2 3 cos3 x − 1 5 cos5 x + C (7.15f)<br />
When there are purely even powers, the substitutions (7.6) and (7.7) reduce<br />
the power of the exponent. In the following example, we have to use the<br />
substitution twice. The first time, we substitute sin 2 x = (1/2)(1−cos(2x)).<br />
The second time, we use cos 2 (2x) = (1/2)(1 + cos(4x))<br />
∫<br />
Example 7.4 sin 4 x dx<br />
∫<br />
∫<br />
sin 4 x dx =<br />
(sin 2 x) 2 dx<br />
(7.16a)<br />
∫ ( ) 2 1<br />
=<br />
2 (1 − cos 2x) dx (7.16b)<br />
= 1 ∫<br />
(1 − 2 cos(2x) + cos 2 (2x)) dx (7.16c)<br />
4<br />
= 1 4 dx − 1 ∫<br />
cos(2x) dx + 1 ∫<br />
cos 2 (2x)dx (7.16d)<br />
2<br />
4<br />
= x 4 − 1 4 sin 2x + 1 ∫<br />
(1 + cos(4x))dx (7.16e)<br />
8<br />
= x 4 − 1 4 sin 2x + x 8 + 1 sin(4x) + C<br />
32<br />
(7.16f)<br />
= 3x 8 − 1 4 sin 2x + 1 sin(4x) + C<br />
32<br />
(7.16g)<br />
When there are a combination of sines and cosines we often have to use a<br />
mix of these rules.<br />
« 2012. Last revised: February 26, 2013 Page 23
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