592
Chapter 5 | Integration
5.5 EXERCISES 254. Why is u -substitution referred to as change of variable ? 255. 2. If f = g ∘ h , when reversing the chain rule, d dx ( g ∘ h )( x ) = g ′( h ( x ) ⎞ ⎠ h ′( x ), should you take u = g ( x ) or u = h ( x )? In the following exercises, verify each identity using differentiation. Then, using the indicated u -substitution, identify f such that the integral takes the form ∫ f ( u ) du .
267. ⌠
3 dx ; u = x 2 −2 x
⌡ ( x −1) ⎛
⎝ x 2 −2 x ⎞ ⎠
268. ⌠ ⌡ ⎛
2 dx ; u = x 3 – 3 x 2
⎝ x 2 −2 x ⎞ ⎠ ⎛
⎝ x 3 −3 x 2 ⎞ ⎠
269. ∫ cos 3 θdθ ; u = sin θ ( Hint : cos 2 θ =1−sin 2 θ )
270. ∫ sin 3 θdθ ; u =cos θ ( Hint : sin 2 θ =1−cos 2 θ )
256. ∫ x x +1 dx = 2 15
( x +1) 3/2 (3 x −2)+ C ; u = x +1
In the following exercises, use a suitable change of variables to determine the indefinite integral. 271. ∫ x (1− x ) 99 dx 272. ⌠ ⌡ t ⎛ ⎝ 1− t 2 ⎞ ⎠ 10 dt 273. ∫ (11 x −7) −3 dx 274. ∫ (7 x −11) 4 dx 275. ∫ cos 3 θ sin θdθ 276. ∫ sin 7 θ cos θdθ 277. ∫ cos 2 ( πt )sin( πt ) dt 278. ∫ sin 2 x cos 3 xdx ( Hint : sin 2 x +cos 2 x =1) 279. ∫ t sin ⎛ ⎝ t 2 ⎞ ⎠ cos ⎛ ⎝ t 2 ⎞ ⎠ dt 280. ∫ t 2 cos 2 ⎛ ⎝ t 3 ⎞ ⎠ sin ⎛ ⎝ t 3 ⎞ ⎠ dt
257.
For
x >1 : ⌠ ⌡
x 2 x −1
⎛ ⎝ 3 x 2 +4 x +8 ⎞
dx = 2 15
x −1
⎠ + C ; u = x −1
258. ⌠ ⌡ x 4 x 2 +9 dx = 1 12 ⎛
3/2
⎝ 4 x 2 +9 ⎞ ⎠
+ C ; u =4 x 2 +9
259. ⌠ ⌡ 260. ⌠ ⌡
x 4 x 2 +9
x 2 +9+ C ; u =4 x 2 +9
dx = 1
4 4
x (4 x 2 +9) 2
; u =4 x 2 +9
dx = − 1
8(4 x 2 +9)
In the following exercises, find the antiderivative using the indicated substitution.
261. ∫ ( x +1) 4 dx ; u = x +1 262. ∫ ( x −1) 5 dx ; u = x −1 263. ∫ (2 x −3) −7 dx ; u =2 x −3 264. ∫ (3 x −2) −11 dx ; u =3 x −2 265. ⌠ ⌡ x x 2 +1 dx ; u = x 2 +1 266. ⌠ ⌡ x 1− x 2 dx ; u =1− x 2
⌠ ⌡ ⎮ x 2 ⎛
dx
281.
2
⎞ ⎠
⎝ x 3 −3
282. ⌠ ⌡
x 3 1− x 2
dx
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online