Jo 17. cos² x sin? x dx

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Can slove 17 and 18 by this method?

 

EXAMPLE 3.4
An Integrand with an Even Power of Sine
Evaluate f sinx dx.
Solution Using the half-angle formula, we can rewrite the integral as
sin x dx =
1- cos 2x) dx.
We can evaluate this last integral by using the substitution u = 2x, so that du = 2 dx.
This gives us
sin?
(1 – cos 2x)2 dx =
(1 – cos u) du
1- cos u
du
- sin u) +c =
(2x – sin 2x) + c.
Since u 2.x.
With some integrals, you may need to apply the half-angle formulas several times, as
in example 3.5.
EXAMPLE 3.5
An Integrand with an Even Power of Cosine
Evaluate f cosx dx.
Solution Using the half-angle formula for cosine, we have
小
1
(1+ cos 2x)? dx
cos x dx =
(cos x) dx =
(1+ 2 cos 2x + cos 2x) dx.
=
Using the half-angle formula again, on the last term in the integrand, we get
| cos“.
dx =
1+2 cos 2x + =(1+ cos 4x) dx
3
1
sin 2x +
sin 4x + c,
32
Transcribed Image Text:EXAMPLE 3.4 An Integrand with an Even Power of Sine Evaluate f sinx dx. Solution Using the half-angle formula, we can rewrite the integral as sin x dx = 1- cos 2x) dx. We can evaluate this last integral by using the substitution u = 2x, so that du = 2 dx. This gives us sin? (1 – cos 2x)2 dx = (1 – cos u) du 1- cos u du - sin u) +c = (2x – sin 2x) + c. Since u 2.x. With some integrals, you may need to apply the half-angle formulas several times, as in example 3.5. EXAMPLE 3.5 An Integrand with an Even Power of Cosine Evaluate f cosx dx. Solution Using the half-angle formula for cosine, we have 小 1 (1+ cos 2x)? dx cos x dx = (cos x) dx = (1+ 2 cos 2x + cos 2x) dx. = Using the half-angle formula again, on the last term in the integrand, we get | cos“. dx = 1+2 cos 2x + =(1+ cos 4x) dx 3 1 sin 2x + sin 4x + c, 32
Jo
J-1/4
| cos x
18. (cos
17.
sin? x dx
x + sin x) dx
7/2
19.
cos x sin' x dx
20.
cot x csc* x dx
-1/3
T/4
1
21.
dx
22.
dx
r2,9
r216
Transcribed Image Text:Jo J-1/4 | cos x 18. (cos 17. sin? x dx x + sin x) dx 7/2 19. cos x sin' x dx 20. cot x csc* x dx -1/3 T/4 1 21. dx 22. dx r2,9 r216
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