606
Chapter 5 | Integration
368. [T] y = ln( x ) over [1, 2]
352. ∫ ln(cos x )tan xdx 353. ∫ ln(csc x )cot xdx 354. ⌠ ⌡ e x − e − x e x + e − x dx
x +1 x 2 +2 x +6
over [0, 1]
369. [T] y =
370. [T] y =2 x over [−1, 0] 371. [T] y =−2 − x over [0, 1]
In the following exercises, evaluate the definite integral. 355. ⌠ ⌡ 1 2 1+2 x + x 2 3 x +3 x 2 + x 3 dx 356. ∫ 0 π /4 tan xdx 357. ⌠ ⌡ 0 π /3 sin x −cos x sin x +cos x dx
In the following exercises, f ( x ) ≥0 for a ≤ x ≤ b . Find the area under the graph of f ( x ) between the given values a and b by integrating.
log 10 ( x ) x
372. f ( x ) =
; a =10, b =100
log 2 ( x )
373. f ( x ) =
x ; a =32, b =64
374. f ( x ) =2 − x ; a =1, b =2 375. f ( x ) =2 − x ; a =3, b =4
π /2
358. ∫
csc xdx
π /6
376. Find the area under the graph of the function f ( x ) = xe − x 2 between x =0 and x =5. 377. Compute the integral of f ( x ) = xe − x 2 and find the smallest value of N such that the area under the graph f ( x ) = xe − x 2 between x = N and x = N +1 is, at most, 0.01. 378. Find the limit, as N tends to infinity, of the area under the graph of f ( x ) = xe − x 2 between x =0 and x =5. 379. Show that ∫ a b dt t = ∫ 1/ b 1/ a dt t when 0< a ≤ b . 380. Suppose that f ( x ) >0 for all x and that f and g are differentiable. Use the identity f g = e g ln f and the chain rule to find the derivative of f g . 381. Use the previous exercise to find the antiderivative of h ( x ) = x x (1+ln x ) and evaluate ∫ 2 3 x x (1+ln x ) dx . 382. Show that if c >0, then the integral of 1/ x from ac to bc (0< a < b ) is the same as the integral of 1/ x from a to b . The following exercises are intended to derive the fundamental properties of the natural log starting from the
π /3
359. ∫
cot xdx
π /4
In the following exercises, integrate using the indicated substitution. 360. ∫ x x −100 dx ; u = x −100 361. ⌠ ⌡ y −1 y +1 dy ; u = y +1
362. ⌠ ⌡ 363. ⌠ ⌡
1− x 2 3 x − x 3
dx ; u =3 x − x 3
sin x +cos x sin x −cos x
dx ; u = sin x −cos x
364. ∫ e 2 x 1− e 2 x dx ; u = e 2 x 365. ⌠ ⌡ ln( x ) 1−(ln x ) 2
x dx ; u = ln x
In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R 50 and solve for the exact area.
366. [T] y = e x over [0, 1] 367. [T] y = e − x over [0, 1]
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