614
Chapter 5 | Integration
426. ⌠
413. ⌠ ⌡ 414. ⌠ ⌡ 415. ⌠ ⌡ 416. ⌠ ⌡
tan −1 (2 t ) 1+4 t 2
⌡ dt t ⎛
dt
⎝ 1+ln 2 t ⎞ ⎠
⎛ ⎝ t 2
⎞ ⎠
t tan −1
427. ⌠ ⌡ 428. ⌠ ⌡
cos −1 (2 t ) 1−4 t 2 e t cos −1 ⎛
dt
dt
1+ t 4
sec −1 ⎞ ⎠ | t | t 2 −4 ⎛ ⎝ t 2
t ⎞ ⎠
⎝ e
dt
dt
1− e 2 t
t sec −1 ⎞ ⎠ t 2 t 4 −1 ⎛ ⎝ t 2
In the following exercises, compute each definite integral. 429. ⌠ ⌡ 0 1/2 tan ⎛ ⎝ sin −1 t ⎞ ⎠ 1− t 2 dt
dt
In the following exercises, use a calculator to graph the antiderivative ∫ f with C =0 over the given interval ⎡ ⎣ a , b ⎤ ⎦ . Approximate a value of C , if possible, such that adding C to the antiderivative gives the same value as the definite integral F ( x ) = ∫ a x f ( t ) dt . 417. [T] ⌠ ⌡ 1 x x 2 −4 dx over ⎡ ⎣ 2, 6 ⎤ ⎦
430. ⌠
⌡ 1/4 1/2
⎛ ⎝ cos −1 t
⎞ ⎠
tan
dt
1− t 2
431. ⌠ ⌡ 0 432. ⌠ ⌡ 0
1/2
⎛ ⎝ tan −1 t
⎞ ⎠
sin
dt
1+ t 2
1/2
⎛ ⎝ tan −1 t
⎞ ⎠
418. [T] ⌠ ⌡ 419. [T] ⌠ ⌡ 420. [T] ⌠ 421. [T] ⌠ ⌡ 422. [T] ⌠
cos
1 (2 x +2) x
dx over ⎡
⎤ ⎦
⎣ 0, 6
dt
1+ t 2
(sin x + x cos x ) 1+ x 2 sin 2 x
dx over ⎡
⎣ −6, 6 ⎤ ⎦
433. For A >0, compute I ( A ) = ⌠ ⌡ − A A dt 1+ t 2
and
e −2 x 1− e −4 x
⌡ 2
dx over [0, 2]
1 1+ t 2
I ( A ), the area under the graph of
evaluate lim a →∞
on [−∞, ∞]. 434. For 1< B <∞, compute I ( B ) = ⌠ ⌡ 1 B
1 x + x ln 2 x
over [0, 2]
dt t t 2 −1
and
−1 x 1− x 2
⌡ sin
over [−1, 1]
lim B →∞
I ( B ), the area under the graph of
evaluate
1 t t 2 −1
over [1, ∞).
In the following exercises, compute each integral using appropriate substitutions. 423. ⌠ ⌡ e t 1− e 2 t dt 424. ⌠ ⌡ e t 1+ e 2 t dt 425. ⌠ ⌡ dt t 1−ln 2 t
435. Use the substitution u = 2cot x and the identity 1+cot 2 x =csc 2 x to evaluate ⌠ ⌡ dx 1+cos 2 x . ( Hint: Multiply the top and bottom of the integrand by csc 2 x .)
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