Name:
Section:
Trigonometric Integrals
Pythagorean Identity: sin2 x + cos2 x = 1, 1 + tan2 x = sec2 x,
2x
2x
Half-angle formula: sin2 x = 1−cos
and cos2 x = 1+cos
2
2
Evaluate the following trigonometric integrals.
1.
R
sin3 xdx
2.
R
cos3 (20x)dx
3.
R
cos4 2θdθ
1
1 + cot2 x = csc2 x
4.
R
sin3 x cos5 xdx
5.
R
sin3 θ cos−2 θdθ
6.
R
√
cos3 x sin xdx)
7.
R
sin− 2 x cos3 xdx
3
2
8.
R
tan2 xdx
9.
R
6 sec4 xdx
10.
R
cot5 3xdx
11.
R
csc10 x cot xdx
12.
Rπ
4
0
tan3 θ sec2 θdθ
3
Trigonometric substitution
Evaluate the following integrals.
R 10 √
100 − x2 dx
1. 5
3
(36 − 9x2 )− 2 dx
2.
R
3.
R√
64 − x2 dx
4.
R
√ dx
dx
x2 −81
4
5.
R1
6.
R √2
7.
R1
8.
R √13 √
0
√ dx
x2 +16
0
√1
3
0
2
√x
dx
4−x2
√dx
x2 1+x2
x2 + 1dx
5
Name: _________________________________ Section: ____________________ Date: ________________
Section 7.3 Quick Quiz
Answer the following multiple choice questions by circling the correct response.
1.
Which identity is most useful in evaluating
(a) cos 2x = 2 cos2 x – 1
2.
Which identity is most useful in evaluating
Which identity is most useful in evaluating
Which identity is most useful in evaluating
The value of
∫
π
∫ sin
2
∫ sin
3
∫ sec
4
(c) sin2 x + cos2 x = 1
x dx ?
(c) sin2 x + cos2 x = 1
x cos 2 x dx ?
(b) cos 2x = 1 − 2 sin2 x
(a) cot2 x + 1 = csc2 x
5.
x dx ?
(b) cos 2x = 1 − 2 sin2 x
(a) cos 2x = 2 cos2 x – 1
4.
2
(b) cos 2x = 1 − 2 sin2 x
(a) cos 2x = 2 cos2 x – 1
3.
∫ cos
(c) sin2 x + cos2 x = 1
x tan 4 x dx ?
(b) 1 + tan2 x = sec2 x
(c) sin2 x + cos2 x = 1
(b) π/2.
(c) 1.
(b) 1.
(c) π.
cos 2 x dx is
0
(a) π.
6.
The value of
(a) 0.
π /4
∫π
− /4
tan x sec 2 x dx is
Copyright © 2015 Pearson Education, Inc.
Name: _________________________________ Section: ____________________ Date: ________________
Section 7.4 Quick Quiz
Answer the following multiple choice questions by circling the correct response.
1.
The appropriate change of variables for the integral
(a) x = 25 tan θ.
2.
3.
4.
5.
6.
dx
x2 − 4
is
∫
(c) x = 2 sin θ.
dx
9 − x2
is
∫
(c) x = 9 sin θ.
dx
8 + x2
is
∫
(c) x = 2 2 tan θ.
dx
9 − 4 x2
is
(b) x = (2/3) sin θ.
The appropriate change of variables for the integral
(a) x + 1 = 3 tan θ.
∫
(c) x = 5 tan θ.
(b) x = 8 sin θ.
The appropriate change of variables for the integral
(a) x = 3 sin θ.
is
(b) x = 3 sin θ.
The appropriate change of variables for the integral
(a) x = 8 tan θ.
x + 25
(b) x = 4 sec θ.
The appropriate change of variables for the integral
(a) x = 3 sec θ.
dx
2
(b) x = 5 sin θ.
The appropriate change of variables for the integral
(a) x = 2 sec θ.
∫
∫
(c) x = (3/2) sin θ.
dx
x + 2 x + 10
2
is
(b) x = 10 sin θ.
Copyright © 2015 Pearson Education, Inc.
(c) x − 1 = 2 tan θ.