7.2 Trigonometric Integrals
Tues Jan 12
Do Now
Evaluate
sin
x
cos
x
dx
ò
4
HW Review
Integrating Powers of Sine
and Cosine
• 1) Determine which power is odd
• 2) Factor out one power of that trig
• 3) Use the trig identity sin2 x + cos2 x =1
to get rid of all powers of that trig
except one
• 4) Use u-substitution on the OTHER trig
function
Ex 3.2 sinx is odd
• Evaluate
ò (cos x) (sin x) dx
4
3
Ex 3.3 cosx is odd
• Evaluate
ò
sin x (cos x)5 dx
Case 1: When tanx is odd
• 1) Factor out tanxsecx
2
2
• 2) Use the trig identity tan x = sec x -1
to remove all powers of trig (except 1)
• 3) Let u = secx (du = tanxsecx)
• 4) Integrate and replace u
Ex 3.6
• Evaluate
ò tan
3
x sec x dx
3
Case 2: secx is even
• 1) Factor out one factor of sec x
2
2
• 2) Use trig identity sec x = tan x +1
to replace remaining factors
2
• 3) Let u = tanx (du = sec x )
• 4) Integrate and replace u
2
Ex 3.7
• Evaluate
ò tan
2
x sec x dx
4
Other cases
• When we have a case where the
substitution method does not work, we
have several reduction formulas that
can be found on p.410
Ex
• Evaluate
ò sec x dx
Ex
• Evaluate
ò tan
3
x dx
Ex
• Evaluate
ò sin 4x cos3x dx
Closure
• If we use a reduction formula for an
integral that could be evaluated by
substitution, we would get 2 different
(but equivalent) answers. Why?
• HW: p.411 #5 11 27 33 43 47 53 57 63
65