05 - Integration by Parts - Kuta Software

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Kuta Software - Infinite Calculus
Name___________________________________
Integration by Parts
Evaluate each indefinite integral using integration by parts. u and dv are provided.
1)
∫
xe x dx; u = x, dv = e x dx 2)
∫
Date________________ Period____
xcos x dx; u = x, dv = cos x dx
3)
∫
x ⋅ 2 x dx; u = x, dv = 2 x dx 4)
∫
x ln x dx; u = ln x, dv =
x dx
Evaluate each indefinite integral.
5)
∫
xe −x dx 6)
∫
x 2 cos 3x dx
7)
∫
x 2
dx 8)
2x
e
∫
x 2 e 5x dx
9)
∫
ln (x + 3) dx 10)
∫
cos 2x ⋅ e −x dx

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Kuta Software - Infinite Calculus
Name___________________________________
Integration by Parts
Evaluate each indefinite integral using integration by parts. u and dv are provided.
1)
∫
xe x dx; u = x, dv = e x dx
xe x − e x + C
xsin x + cos x + C
2)
∫
Date________________ Period____
xcos x dx; u = x, dv = cos x dx
3)
∫
x ⋅ 2 x dx; u = x, dv = 2 x dx
x ⋅ 2 x
ln 2 −
4)
2 x
(ln 2) + C 2
∫
x ln x dx; u = ln x, dv =
3
2x
2 ln x
3
− 4x 3
2
9 + C
x dx
Evaluate each indefinite integral.
5)
∫
xe −x dx
6)
∫
x 2 cos 3x dx
Use: u = x, dv = e −x dx
∫
xe −x dx = −x − 1
e x
+ C
Use: u = x 2 , dv = cos 3x dx
∫
x 2 cos 3x dx = x2 sin 3x
3
2xcos 3x
+
9
2sin 3x
− + C
27
7)
∫
x 2
dx
2x
e
Use: u = x 2 , dv = 1
∫
x 2
e
2x
dx
e dx = −2x2 − 2x − 1
2x 4e 2x
+ C
8)
∫
x 2 e 5x dx
Use: u = x 2 , dv = e 5x dx
x 2 e 5x dx = x2 e 5x
− 2xe5x
5 25
∫
+ 2e5x
125 + C
9)
∫
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 2x ⋅ e −x dx
Use: u = e −x , dv = cos 2x dx
∫
cos 2x ⋅ e −x 2sin 2x − cos 2x
dx = + C
5e x
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