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ID: 1
AP Calculus AB
Name___________________________________
Integration by Parts
Date________________ Period____
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Evaluate each indefinite integral using integration by parts. u and dv are provided.
1)
3)
∫ ln x dx; u = ln x, dv = dx
∫
ln x
1
dx; u = ln x, dv = 2 dx
2
x
x
2)
∫ xcos x dx; u = x, dv = cos x dx
4)
∫ tan
−1
x dx; u = tan −1 x, dv = dx
Evaluate each indefinite integral.
5)
∫
x 2 ln x dx
6)
∫
7)
∫ x sin x dx
8)
∫ x ⋅ sec x dx
2
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x
xe dx
2
Worksheet by Kuta Software LLC
ID: 1
AP Calculus AB
Name___________________________________
Integration by Parts
Date________________ Period____
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Evaluate each indefinite integral using integration by parts. u and dv are provided.
1)
∫ ln x dx; u = ln x, dv = dx
2)
∫ xcos x dx; u = x, dv = cos x dx
x ln x − x + C
3)
∫
xsin x + cos x + C
ln x
1
dx; u = ln x, dv = 2 dx
2
x
x
4)
∫ tan
− ln x − 1
+C
x
−1
x dx; u = tan −1 x, dv = dx
xtan −1 x −
ln ( x 2 + 1)
+C
2
Evaluate each indefinite integral.
5)
∫
x 2 ln x dx
6)
Use: u = ln x, dv = x 2 dx
x 3 ln x x 3
2
x ln x dx =
−
+C
3
9
x
Use: u = x, dv = e dx
∫
7)
∫ x sin x dx
2
∫
x
xe dx
∫ xe dx = xe − e + C
x
8)
x
x
∫ x ⋅ sec x dx
2
Use: u = x 2 , dv = sin x dx
Use: u = x, dv = sec 2 x dx
∫ x sin x dx = −x cos x + 2xsin x + 2cos x + C
∫ x ⋅ sec x dx = xtan x + ln cos x + C
2
2
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2
Worksheet by Kuta Software LLC