Chapter 5 | Integration
605
5.6 EXERCISES In the following exercises, compute each indefinite integral.
337. ∫ x 2 e − x 3
dx
320. ∫ e 2 x dx 321. ∫ e −3 x dx 322. ∫ 2 x dx 323. ∫ 3 − x dx 324. ⌠ ⌡ 1 2 x dx 325. ∫ 2 x dx 326. ⌠ ⌡ 1 x 2 dx 327. ∫ 1 x dx
338. ∫ e sin x cos xdx 339. ∫ e tan x sec 2 xdx 340. ∫ e ln x dx x 341. ⌠ ⌡ e ln(1− t ) 1− t dt
In the following exercises, verify by differentiation that ∫ ln xdx = x (ln x −1)+ C , then use appropriate changes of variables to compute the integral. 342. ∫ ln xdx ( Hint : ⌠ ⌡ ln xdx = 1 2 ∫ x ln ⎛ ⎝ x 2 ⎞ ⎠ dx ) 343. ∫ x 2 ln 2 xdx 344. ⌠ ⌡ ln x x 2 dx ( Hint :Set u = 1 x .) 345. ∫ ln x x dx ( Hint :Set u = x .) 346. Write an integral to express the area under the graph of y = 1 t from t =1 to e x and evaluate the integral. 347. Write an integral to express the area under the graph of y = e t between t =0 and t = ln x , and evaluate the integral. In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. 348. ∫ tan(2 x ) dx 349. ⌠ ⌡ sin(3 x )−cos(3 x ) sin(3 x )+cos(3 x ) dx
In the following exercises, find each indefinite integral by using appropriate substitutions. 328. ∫ ln x x dx 329. ⌠ ⌡ dx x (ln x ) 2 330. ⌠ ⌡ dx x ln x ( x >1) 331. ⌠ ⌡ dx x ln x ln(ln x ) 332. ∫ tan θdθ 333. ∫ cos x − x sin x x cos x dx 334. ⌠ ⌡ ln(sin x ) tan x dx 335. ∫ ln(cos x )tan xdx 336. ∫ xe − x 2 dx
350. ⌠ ⌡
⎛ ⎝ x 2
⎞ ⎠
x sin
dx
⎛ ⎝ x 2
⎞ ⎠
cos
351. ∫ x csc ⎛
⎞ ⎠ dx
⎝ x 2
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