Trigonometric Integrals - Bruce E. Shapiro
Trigonometric Integrals - Bruce E. Shapiro
Trigonometric Integrals - Bruce E. Shapiro
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Math 150A TOPIC 7. TRIGONOMETRIC INTEGRALS<br />
and an appropriate u-substitution of either u = sec x or u = tan x to<br />
solve the integral. Unfortunately, the process is complicated by the fact<br />
that, unlike in the case of sines and cosines, we do not have double-angle<br />
formulas that reduce the powers. Thus our combination of substitutions<br />
will not always work. It is helpful to recall the basic integration formulas<br />
here as well:<br />
∫<br />
∫<br />
tan x dx = ln | sec x| + C<br />
sec x dx = ln | sec x + tan x| + C<br />
(7.20a)<br />
(7.20b)<br />
∫<br />
Example 7.7<br />
tan 3 x dx<br />
∫<br />
∫<br />
tan 3 x dx =<br />
∫<br />
=<br />
∫<br />
=<br />
tan x tan 2 x dx<br />
tan x(sec 2 x − 1) dx<br />
∫<br />
tan x sec 2 x dx − tan x dx<br />
(7.21a)<br />
(7.21b)<br />
(7.21c)<br />
= 1 2 tan2 xdx − ln | sec x| + C (7.21d)<br />
where we have used u = tan x in the first integral.<br />
∫<br />
Example 7.8 tan 6 x sec 4 x dx<br />
∫<br />
∫<br />
tan 6 x sec 4 x dx =<br />
∫<br />
=<br />
∫<br />
=<br />
tan 6 x sec 2 x sec 2 x dx<br />
tan 6 x(1 + tan 2 x) sec 2 x dx<br />
∫<br />
tan 6 x sec 2 x dx + tan 8 x sec 2 x dx<br />
(7.22a)<br />
(7.22b)<br />
(7.22c)<br />
= 1 7 tan7 x + 1 9 tan9 x + C (7.22d)<br />
where we have used u = tan x in each integral.<br />
Page 26 « 2012. Last revised: February 26, 2013
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