Trigonometric Integrals - Bruce E. Shapiro

Math 150A TOPIC 7. TRIGONOMETRIC INTEGRALS
If we divide equation (7.4) by cos x we get a similar equation relating the
tangent and secant of an angle:
Finally, we add the double angle formula for a sine
tan 2 x + 1 = sec 2 x (7.8)
sin 2x = 2 sin x cos x (7.9)
and the following product-angle formulas to the mix:
sin x cos y = 1 [sin(x − y) + sin(x + y)] (7.10)
2
sin x sin y = 1 [cos(x − y) − cos(x + y)] (7.11)
2
cos x cos y = 1 [cos(x − y) + cos(x + y)] (7.12)
2
Equations (7.4) through (7.12) will be our main tools for finding substitutions
that reduce integrals to manageable proportions.
∫
Example 7.1 sin 2 x dx This is perhaps the most basic of the integrals
of this form. We can use the identity (7.7) to obtain
∫
sin 2 x dx = 1 ∫
(1 − cos 2x) dx (7.13a)
2
= 1 2 x − 1 sin 2x + C (7.13b)
4
When there is a single odd power (e.g., cos 3 x, sin 3 x, cos 5 x, sin 5 x, . . . ), we
“reserve” one factor for the derivative, and replace all the other factors with
the identity (7.4), followed by a u-substitution of u = cos x or u = sin x.
∫
Example 7.2 cos 3 x dx
∫
∫
cos 3 x dx = cos x · cos 2 x dx
(7.14a)
∫
= cos x (1 − sin 2 x) dx
(7.14b)
∫
= (1 − u 2 ) du where u = sin x (7.14c)
= u − 1 3 u3 + C (7.14d)
= sin x − 1 3 sin3 x + C (7.14e)
Page 22 « 2012. Last revised: February 26, 2013