Download 7.3 Trigonometric Integrals

Pop Quiz
1. Give the formula for integration by parts
2. Identify u and dv for integration by parts
of:
4x
A)  x dx
B)  6 arccos xdx
e
4
x
3. Integrate  ln xdx
Trigonometric Integrals
Section 7.3 AP Calc
Identities:
sin 2 x  cos 2 x  1
1  cos 2 x
sin x 
2
2
1  cos 2 x
cos x 
2
2
Guidelines for Sine and Cosine
1) If power of sinx odd and positive, save
one sinx, convert others to cosx
2) If power of cosx is odd and positive, save
one cosx, convert others to to sinx
3) If both sinx and cosx are even and
positive, use half angle identities to
convert to odd power of cosx, then use #2
Solve:
 sin
3
x cos xdx
 cos
2
x sin xdx
Solve:
 sin
4
3
x cos xdx
Solve:
 sin
5
4
x cos xdx
Solve:

5
sin x
dx
cos x
Solve:
 sin
2
(2 x)dx
Wallis’s Formula

 2  4  6   n  1 
n
2
If n odd (n≥3), then 0 cos xdx     ....

 3  5  7   n 

If n even (n≥2), then 02 cosn xdx   1  3  5 .... n  1   
 2  4  6   n  2 
Using Wallis’ formula find the sum:


2
0
5
sin xdx
Identities:
1  tan x  sec x
2
2
Guidelines for Secant and Tangent
1) If power of secx even and positive, save
one sec²x, convert others to tanx
2) If power of tanx is odd and positive, save
secxtanx, convert other tanx to secx
Secant, Tangent continued:
3) If no secx, and tanx is even and positive,
convert tan²x to (sec²x -1), expand, repeat
4) If only secmx, m odd and positive, use
integration by parts
5) If none of first four guidelines apply,
convert to sinx and cosx
Solve:
 sec
2
x tan x dx
Solve:
 tan
3
x sec xdx
Solve:
 tan
6
xdx
Integrals with sine and cosine products with
different angles:
sin( mx) sin( nx)  12 cos(m  n) x  cos(m  n) x
sin( mx) cos(nx)  12 sin (m  n) x  sin (m  n) x
cos(mx) cos(nx)  12 cos(m  n) x  cos(m  n) x
Evaluate
 cos(6 x) cos(2 x)dx