January 17, 2017
WARMUP! Let's review our identities!
The following identities will be needed in order
to be successful with this section. See how
many you can fill in!
1) sin2x = 1 - cos2x
2) cos2x = 1 - sin2x
3) sin2x = 2sinxcosx
2
2
4) cos2x = cos x - sin x
2
cos2x = 2cos x - 1
cos2x = 1 - 2sin2x
Based on the cos2x identities,
5) sin2x = 1/2 (1 - cos2x)
6) cos2x = 1/2 (1 + cos2x)
2
5) tan2x = sec x - 1
6) sec2x = tan2x + 1
January 17, 2017
January 17, 2017
January 17, 2017
January 17, 2017
8.3a Trig Integrals!!
Essential Learning Target:
• Techniques for finding antiderivatives
include algebraic manipulation such as long
division, completing the square, substitution
of variables and integration by parts
January 17, 2017
Diving in!
If one of the exponents is odd, that's the
function you want to convert. Always save one
extra for your u-sub.
(If they're both odd, then it doesn't matter which
one you pick.) (If they're both even, we'll
discuss that later.)
∫ sin5xcos4xdx =
1) pull one sinx out to use later on
2) rewrite the sin4x as (1 - cos2x)2
3) square the binomial
4) distribute the cosines
5) let u = cosx and work from there
January 17, 2017
ex) ∫sin3xcos3xdx
January 17, 2017
You try!
∫sin2(4x)cos3(4x)dx
January 17, 2017
Try again!
∫sin3xdx
January 17, 2017
What if both exponents are even?
Then you need to use the other identities to
convert a function to an odd power of cosine.
ex) ∫sin2xcos4xdx
1) rewrite sin2x as 1/2 (1 - cos2x)
2) rewrite cos4x as (1/2 (1 + cos2x))2
3) multiply out the squared binomial
4) distribute everything and simplify
5) reduce the squared term and integrate as
much as possible
6) integrate the cubed term as you would any
odd-powered trig function by pulling out one
term, converting the other function and using usub
January 17, 2017
ex) ∫sin4(5x)dx
1) rewrite sin4(5x) as (1/2 (1 - cos(10x)))2
2) multiply out the binomial
3) rewrite the cos2(10x) as 1/2 (1 + cos(20x))
4) integrate!
January 17, 2017
That's great and all, but what about tangent?
Because tan2x = sec2x - 1 and the derivative of tanx
is sec2x, it's useful to take advantage of the link
If the power of secx is even, save a sec2x for the du
and convert the rest to tangents
ex) ∫sec4(3x)tan3(3x)dx
1) split up the sec4(3x) and convert one of the
squares to tangents
2) distribute everything
3) u-sub and integrate
January 17, 2017
January 17, 2017
You try!
∫tan4(2x)dx
1) split up the tangent into 2 tangent squares
2) convert one of the tan2(2x) to secant squared
3) distribute everything
4) for the tan2(2x) that is left over, convert it to a
secant squared
5) integrate!
January 17, 2017
Let's sum it up!
• odd powers of sine and/or cosine - pull out one function
for your du and convert the rest to the same function,
then let u = that function
• even powers of sine and/or cosine - use reduction
formula(s) to reduce to an odd power - distribute
everything and continue the process with any
remaining even powers until everything is either to the
first power (integrate) or an odd power (use odd power
method above)
• for secants and tangents, if the power of secant is
even, pull a squared out for du, convert everything else
to tangents, and u-sub
• if your integral is just secant to an odd power, use funky
integration by parts
January 17, 2017
What have we learned?
• Can I evaluate integrals of a variety of
functions involving powers of sine, cosine,
secant and tangent?