Download Trigonometric Integral

Trigonometric Integral
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 7.3 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource)
and your lecture notes.
EXPECTED SKILLS:
• Know antiderivatives for all six elementary trigonometric functions.
• Be able to evaluate integrals that involve powers of sine, cosine, tangent, and secant
by using appropriate trigonometric identities.
PRACTICE PROBLEMS:
1. Fill in the following table
Z
sin x dx =
Z
cos x dx =
Z
tan x dx =
Z
cot x dx =
Z
sec x dx =
Z
csc x dx =
Z
π/3
cot 2xdx
2.
π/4
Powers of Sines & Cosines: For each of the following, evaluate the given integral.
Z
3.
sin (x) cos3 (x) dx
Z
4.
sin3 (x) cos4 (x) dx
Z √
sin x cos3 (x) dx
5.
1
Z
6.
Z
7.
Z
8.
Z
sin2 x dx
sin3 (bx) dx, where b is a non-zero constant
sin2 x cos2 x dx
π/2
cos3 x dx
9.
π/4
Z
10.
cos4 5x dx
11. Consider the trigonmetric identity sin (A + B) = sin A cos B + cos A sin B
(a) Use this identity to derive an identity for sin (A − B) in terms of sin A, cos A,
sin B, and cos B.
(b) Use the given identity and your answer for part (a) to derive the following identity:
sin A cos B =
1
[sin (A − B) + sin (A + B)]
2
12. Consider the trigonmetric identity cos (A + B) = cos A cos B − sin A sin B
(a) Use this identity to derive an identity for cos (A − B) in terms of sin A, cos A,
sin B, and cos B.
(b) Use the given identity and your answer for part (a) to derive the following identity:
cos A cos B =
1
[cos (A − B) + cos (A + B)]
2
(c) Use the given identity and your answer for part (a) to derive the following identity:
sin A sin B =
1
[cos (A − B) − cos (A + B)]
2
For problems 13-16, use an appropriate identity from problem 11 or 12 to evaluate the given integral.
Z
x
13.
sin (2x) cos
dx
2
Z
14.
cos (3x) cos (4x) dx
Z
15.
sin (5x) cos (2x) dx
2
16. The graph of f (x) = sin 2x sin 5x on the interval [−π, π] is shown below.
Compute the net signed area between the graph of f (x) and the x-axis on the interval
[−π, π]
Powers of Tangents & Secants: For each of the following, evaluate the given integral.
Z
17.
tan2 3x dx
Z
18.
π/4
tan3 (x) sec3 (x) dx
0
Z
19.
Z
20.
Z
21.
Z
22.
Z
23.
tan (x) sec3 (x)dx
tan3 (x) sec4 (x) dx
tan5 (2x) sec2 (2x) dx
tan (x) sec5/2 (x) dx
sec4 x dx
Z
π
24. Consider
sec x dx
π/2
(a) Explain why this integral is improper.
(b) Evaluate the given integral. If it diverges, explain why.
3
Z
25. (a) Use integration by parts to evaluate
sec3 (x) dx. (Hint: sec3 x = sec2 x sec x
and tan2 x = sec2 x − 1)
Z
(b) Use part (a) to evaluate
tan2 (x) sec (x) dx
26. Let R beh the iregion bounded between the graphs of y = sin x and y = cos x on the
π π
, .
interval
4 2
(a) Compute the area of R.
(b) Compute the volume of the solid which results from revolving R around the x-axis.
π 3π
27. Find the length of the curve y = ln (sin x) on the interval
,
.
4 4
4