Download An example from the last lecture Evaluate Z sec

An example from the last lecture
Evaluate
Z
π
4
sec3 x dx.
0
A. (ln |secx + tan x|)3 + C
√
√
B. 22 + 12 ln( 2 + 1)
√
3
C. ln( 2 + 1)
D.
π
4
Remark: Use integration by parts.
Math 105 (Section 204)
Integrals via trigonometric substitution
2011W T2
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Identifying a viable strategy
Which of the following strategies would you apply to compute the
following indefinite integral?
Z
sec3 x tan x dx.
A. integration by parts with u 0 = sec 2 x, v = sec x tan x
B. integration by parts with u 0 = tan x, v = sec3 x
C. substituting u = tan x
D. substituting u = sec x
Remark: Refer to the strategies outlined in Tables 7.1 and 7.2 in
section 7.2.
Math 105 (Section 204)
Integrals via trigonometric substitution
2011W T2
2/5
Example : secm x tann x; m, n odd
Find f ( π4 ) if
f 0 (x) = sec3 x tan x,
A.
√
2
3(
1
f (0) = .
3
2 + 1)
B. undefined
C. 1
D. 2/3
Math 105 (Section 204)
Integrals via trigonometric substitution
2011W T2
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Trigonometric substitutions
If your integral contains
a2 − x 2
x 2 + a2
x 2 − a2
Math 105 (Section 204)
substitute
x = a sin θ
x = a tan θ
x = a sec θ
use the identity
a2 − a2 sin2 θ = a2 cos2 θ
a2 + a2 tan2 θ = a2 sec2 θ
a2 sec2 θ − a2 = a2 tan2 θ
Integrals via trigonometric substitution
2011W T2
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Example
Find the value of the integral
Z
0
2
√
x2
dx.
4 + x2
√
A. 2 2
B. −3
√
√
C. 2 2 − 2 ln( 2 + 1)
D. ln 2
Remark: Use an appropriate substitution as outlined in Table 7.3.
Math 105 (Section 204)
Integrals via trigonometric substitution
2011W T2
5/5