substitution. Sketch and label the associated dr 1. Í -2 sine

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Solve 1,3,5
30. COs de
6:05
O X all 66%i
View embed
IMG_5799.png
SECTION 7.3 Trigonometric Substitution
491
We now substitute u= 2 sin 8, giving du = 2 cos 8 de and 4 - u = 2 cos 6, so
Figure 5 shows the graphs of the inte-
grand in Example 7 and its indefinite
integral (with C- 0). Which is which?
2 sin e - 1
2 cos e
dx-
2 cos e de
2x
- (2 sin - 1) de
--2 cos 0 - 0 +C
= -4 - - sin
+ C
FIGURE S
--3- 2x - x - sin
+ C
7.3 EXERCISES
17. Í*
dx
1-3 Evaluate the integral using the indicated trigonometric
substitution. Sketch and label the associated right triangle.
dx
18.
[lar)' - 6']
1.A
dx
1. E
I-2 sine
20. Íd
19.
dx
2.
V+ 4
*-2 tan
21. "-
dx
v9- 25x
22. Va+i de
V -4
3.
- 2 sec
23. j-
24. V - dx
+ 2x + 5
25. j+ 2x – x" de
26.a+ 4x - 4x
4-30 Evaluate the integral.
dx
4. Íd
27. j va + 2x de
19
28.
(x - 2x + 2)
5. j
V -1
de
6.
v36
dx
29. xT -x" dx
cos r
30. .
VI + sin'r
7. ya a>0
8.
a. - 16
dx
31. (a) Use trigonometric substitution to show that
9. "y
10. V4 - 9x ds
- In(x + V + a) + c
dr
11. ("- 4x dx
12.
v4 +
(b) Use the hyperbolic substitution - a sinh to show that
13. j - de
14. d
- sinh
+C
16. (
dx
These formulas are connected by Formula 3.11.3.
E sCm lea AR kd May d
d ti
ni
e a
O
く
Transcribed Image Text:30. COs de 6:05 O X all 66%i View embed IMG_5799.png SECTION 7.3 Trigonometric Substitution 491 We now substitute u= 2 sin 8, giving du = 2 cos 8 de and 4 - u = 2 cos 6, so Figure 5 shows the graphs of the inte- grand in Example 7 and its indefinite integral (with C- 0). Which is which? 2 sin e - 1 2 cos e dx- 2 cos e de 2x - (2 sin - 1) de --2 cos 0 - 0 +C = -4 - - sin + C FIGURE S --3- 2x - x - sin + C 7.3 EXERCISES 17. Í* dx 1-3 Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. dx 18. [lar)' - 6'] 1.A dx 1. E I-2 sine 20. Íd 19. dx 2. V+ 4 *-2 tan 21. "- dx v9- 25x 22. Va+i de V -4 3. - 2 sec 23. j- 24. V - dx + 2x + 5 25. j+ 2x – x" de 26.a+ 4x - 4x 4-30 Evaluate the integral. dx 4. Íd 27. j va + 2x de 19 28. (x - 2x + 2) 5. j V -1 de 6. v36 dx 29. xT -x" dx cos r 30. . VI + sin'r 7. ya a>0 8. a. - 16 dx 31. (a) Use trigonometric substitution to show that 9. "y 10. V4 - 9x ds - In(x + V + a) + c dr 11. ("- 4x dx 12. v4 + (b) Use the hyperbolic substitution - a sinh to show that 13. j - de 14. d - sinh +C 16. ( dx These formulas are connected by Formula 3.11.3. E sCm lea AR kd May d d ti ni e a O く
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