05 - Integration By Parts.pdf 6e6p1p

Kuta Software - Infinite Calculus

Name___________________________________

Integration by Parts

Date________________ Period____

Evaluate each indefinite integral using integration by parts. u and dv are provided. 1)

∫ xe dx; u = x, dv = e dx

2)

∫ xcos x dx; u = x, dv = cos x dx

3)

∫

4)

∫

dx

6)

∫ x cos 3x dx

dx

8)

∫xe

x

x

x

x

x ⋅ 2 dx; u = x, dv = 2 dx

x ln x dx; u = ln x, dv =

x dx

Evaluate each indefinite integral. 5)

∫ xe

7)

∫

9)

∫

−x

x2 e

2x

ln ( x + 3) dx

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10)

2

2 5x

∫

dx

cos 2 x ⋅ e − x dx

Worksheet by Kuta Software LLC

Kuta Software - Infinite Calculus

Name___________________________________

Integration by Parts

Date________________ Period____

Evaluate each indefinite integral using integration by parts. u and dv are provided. 1)

∫ xe dx; u = x, dv = e dx x

x

x

2)

∫ xcos x dx; u = x, dv = cos x dx

x

xsin x + cos x + C

xe − e + C

3)

∫

x

x

x ⋅ 2 dx; u = x, dv = 2 dx x

4)

∫

x ln x dx; u = ln x, dv = 3 2

x

x⋅2 2 − +C ln 2 (ln 2) 2

x dx

3 2

2 x ln x 4 x − +C 9 3

Evaluate each indefinite integral. 5)

∫ xe

−x

6)

dx

Use: u = x, dv = e − x dx −x − 1 −x xe dx = +C x e

∫

x2 e

8)

dx

2

∫ 9)

∫

∫

2x

Use: u = x , dv = x2 e 2x

dx =

1

∫xe

2 5x

dx 2

5x

Use: u = x , dv = e dx x2 e5x 2 xe 5 x 2e 5 x 2 5x x e dx = − + +C 5 25 125

dx

e 2x −2 x 2 − 2 x − 1 4e 2 x

2

Use: u = x 2 , dv = cos 3 x dx x 2 sin 3 x 2 xcos 3 x 2sin 3 x 2 x cos 3 x dx = + − +C 3 9 27

∫

7)

∫ x cos 3x dx

∫

+C

ln ( x + 3) dx Use: u = ln ( x + 3) , dv = dx *or use u-subs first

∫ ln (x + 3) dx = x ln (x + 3) − x + 3 ln (x + 3) + C

10)

∫

cos 2 x ⋅ e − x dx

Use: u = e − x, dv = cos 2 x dx 2sin 2 x − cos 2 x −x cos 2 x ⋅ e dx = +C x 5e

∫

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Worksheet by Kuta Software LLC