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 =
− cos x + C
cos x dx =
sin x + C
tan x dx =
ln | sec x| + C
cot x dx =
ln | sin x| + C
sec x dx =
ln | sec x + tan x| + C
csc x dx =
− ln | csc x + cot x| + C
Z
Z
Z
Z
Z
Z
π/3
2.
cot 2xdx
π/4
1
1
ln 3 − ln 2
4
2
Powers of Sines & Cosines: For each of the following, evaluate the given integral.
Z
3.
sin (x) cos3 (x) dx
1
− cos4 x + C
4
1
Z
4.
sin3 (x) cos4 (x) dx
1
1
cos7 x − cos5 x + C
7
5
Z √
5.
sin x cos3 (x) dx
2
2
(sin x)3/2 − (sin x)7/2 + C
3
7
Z
6.
sin2 x dx
x 1
− sin (2x) + C
2 4
Z
7.
sin3 (bx) dx, where b is a non-zero constant
1
1
cos3 (bx) − cos (bx) + C
3b
b
Z
8.
sin2 x cos2 x dx
x
1
−
sin (4x) + C
8 32
Z π/2
9.
cos3 x dx
π/4
√
2 5 2
−
3
12
Z
10.
cos4 5x dx
3
1
1
x+
sin (10x) +
sin (20x) + C
8
20
160
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.
sin (A − B) = sin A cos B − cos A sin B
2
(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
Adding the given identity to the identity from part (a) and then dividing both
sides by 2 yields the desired result.
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.
cos (A − B) = cos A cos B + sin A sin 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
Adding the given identity to the identity from part (a) and then dividing both
sides by 2 yields the desired result.
(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
Subtracting the given identity from the identity from part (a) and then dividing
both sides by 2 yields the desired result.
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
1
5x
1
3x
− cos
− cos
+C
5
2
3
2
Z
14.
cos (3x) cos (4x) dx
1
1
sin x +
sin (7x) + C
2
14
Z
15.
sin (5x) cos (2x) dx
1
1
− cos (3x) −
cos (7x) + C
6
14
3
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]
0
Powers of Tangents & Secants: For each of the following, evaluate the given integral.
Z
17.
tan2 3x dx
−x +
Z
18.
1
tan (3x) + C
3
π/4
tan3 (x) sec3 (x) dx
0
√ 2 1+ 2
15
Z
19.
tan (x) sec3 (x)dx
1
sec3 x + C
3
Z
20.
tan3 (x) sec4 (x) dx
1
1
tan6 x + tan4 x + C
6
4
Z
21.
tan5 (2x) sec2 (2x) dx
1
tan6 (2x) + C
12
4
Z
22.
tan (x) sec5/2 (x) dx
2
sec5/2 x + C
5
Z
23.
sec4 x dx
1
tan3 x + tan x + C
3
Z π
24. Consider
sec x dx
π/2
(a) Explain why this integral is improper.
π
The integral is improper because sec x has an infinite discontinuity at x =
2
which is the lower limit of integration.
(b) Evaluate the given integral. If it diverges, explain why.
Z π
sec x dx = −∞
The integral diverges because
π/2
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)
1
1
sec (x) tan (x) + ln | sec x + tan x| + C
2
2
Z
(b) Use part (a) to evaluate tan2 (x) sec (x) dx
1
1
sec (x) tan (x) − ln | sec x + tan x| + C
2
2
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.
√
2−1
(b) Compute the volume of the solid which results from revolving R around the x-axis.
π
2
5
π 3π
27. Find the length of the curve y = ln (sin x) on the interval
,
.
4 4
√ 2 ln 2 + 2 − ln 2
6