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Chapter 6 | Applications of Integration
6.7 EXERCISES
For the following exercises, find the derivative dy dx . 295. y = ln(2 x ) 296. y = ln(2 x +1) 297. y = 1 ln x For the following exercises, find the indefinite integral. 298. ∫ dt 3 t 299. ∫ dx 1+ x For the following exercises, find the derivative dy / dx . (You can use a calculator to plot the function and the derivative to confirm that it is correct.)
2
312. ∫
xdx x 2 +1 x 3 dx x 2 +1
0
2
313. ∫
0
e
314. ∫
dx x ln x
2
e
315. ∫ dx x (ln x ) 2 316. ∫ cos xdx sin x 317. ∫ 0 π /4 tan xdx 318. ∫ cot(3 x ) dx 319. ∫ (ln x ) 2 dx x For the following exercises, compute dy / dx by differentiating ln y . 320. y = x 2 +1 321. y = x 2 +1 x 2 −1 2
x )
300. [T] y = ln( x 301. [T] y = x ln( x ) 302. [T] y = log 10 x 303. [T] y = ln(sin x ) 304. [T] y = ln(ln x ) 305. [T] y =7 ln(4 x )
322. y = e sin x 323. y = x −1/ x 324. y = e ( ex )
306. [T] y = ln ⎛
⎞ ⎠
⎝ (4 x ) 7
307. [T] y = ln(tan x ) 308. [T] y = ln(tan(3 x ))
325. y = x e 326. y = x (
309. [T] y = ln ⎛
⎞ ⎠
⎝ cos 2 x
ex )
For the following exercises, find the definite or indefinite integral. 310. ∫ 0 1 dx 3+ x 311. ∫ 0 1 dt 3+2 t
327. y = x x 3 x 6 328. y = x −1/ln x 329. y = e −ln x
For the following exercises, evaluate by any method.
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