Chapter 5 | Integration
613
5.7 EXERCISES In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 391. ⌠ ⌡ 0 3/2 dx 1− x 2 392. ⌠ ⌡ −1/2 1/2 dx 1− x 2 393. ⌠ ⌡ 3 1 dx 1 - x 2 394. ⌠ ⌡ 1/ 3 3 dx 1+ x 2 395. ⌠ ⌡ 1 2 dx | x | x 2 −1 396. ⌠ ⌡ 1 2/ 3 dx | x | x 2 −1 In the following exercises, find each indefinite integral, using appropriate substitutions. 397. ⌠ ⌡ dx 9− x 2 398. ⌠ ⌡ dx 1−16 x 2 399. ⌠ ⌡ dx 9+ x 2 400. ⌠ ⌡ dx 25+16 x 2 401. ⌠ ⌡ dx | x | x 2 −9 402. ⌠ ⌡ dx | x | 4 x 2 −16
403.
Explain
the
relationship
−cos −1 t + C = ⌠ ⌡
dt 1− t 2
= sin −1 t + C . Is it true, in
general, that cos −1 t =−sin −1 t ? 404. Explain the
relationship
sec −1 t + C = ⌠
⌡ dt
=−csc −1 t + C . Is it true, in
| t | t 2 −1
general, that sec −1 t =−csc −1 t ? 405. Explain what is wrong with the following integral: ⌠ ⌡ 1 2 dt 1− t 2 . 406. Explain what is wrong with the following integral: ⌠ ⌡ −1 1 dt | t | t 2 −1 . In the following exercises, solve for the antiderivative ∫ f of f with C =0, then use a calculator to graph f and the antiderivative over the given interval ⎡ ⎣ a , b ⎤ ⎦ . Identify a value of C such that adding C to the antiderivative recovers the definite integral F ( x ) = ∫ a x f ( t ) dt . 407. [T] ⌠ ⌡ 1 9− x 2 dx over [−3, 3] 408. [T] ⌠ ⌡ 9 9+ x 2 dx over ⎡ ⎣ −6, 6 ⎤ ⎦
409. [T] ⌠ ⌡ 410. [T] ⌠ ⌡
cos x 4+sin 2 x
dx over ⎡
⎣ −6, 6 ⎤ ⎦
e x 1+ e 2 x
dx over ⎡
⎣ −6, 6 ⎤ ⎦
In the following exercises, compute the antiderivative using appropriate substitutions. 411. ⌠ ⌡ sin −1 tdt 1− t 2 412. ⌠ ⌡ dt sin −1 t 1− t 2
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