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Integration by Parts (Examples)

Integration of the product of an exponential and polynomial

The evaluation of the integral

(1) ⌠⌡x²e^xdx

can be performed using integration by parts. By invoking the substitutions:

(2) f = x² ⇒ df = 2xdx dg = e^xdx ⇒ g = e^x

Then the integration becomes:

(3) ⌠⌡x²e^xdx = x²e^x − ⌠⌡2xe^xdx + C

where C is a constant of integration.

Utilizing integration by parts once more allows us to reduce the integral (second term on the right hand side in Equation 3↑):

(4) f = 2x ⇒ df = 2dx dg = e^xdx ⇒ g = e^x

The second term in Equation 3↑ reduces to:

(5) ⌠⌡2xe^xdx = 2xe^x − ⌠⌡2e^xdx + D = 2xe^x − 2e^x

where D is a constant of integration.

The entire expression then amounts to:

(6) ⌠⌡x²e^xdx = x²e^x − 2xe^x + 2e^x + C + D

Integration of the product of an exponential and a trigonometric function

The evaluation of the integral

(7) ⌠⌡e^xcos(x)dx

can be performed using integration by parts. By invoking the substitutions:

(8) f = cos(x) ⇒ df = − sin(x)dx dg = e^xdx ⇒ g = e^x

(9) ⌠⌡e^xcos(x)dx = e^xcos(x) + ⌠⌡e^xsin(x)dx + C

where C is a constant of integration.

The second term of Equation 9↑ can be integrated again using:

(10) f = sin(x) ⇒ df = cos(x)dx dg = e^xdx ⇒ g = e^x

(11) ⌠⌡e^xsin(x)dx = e^xsin(x) − ⌠⌡e^xcos(x)dx + D

where D is a constant of integration.

Recognizing that the second term of Equation 11↑ is the original expression, we can get:

(12) ⌠⌡e^xcos(x)dx = e^xcos(x) + e^xsin(x) − ⌠⌡e^xcos(x)dx + C + D ⌠⌡e^xcos(x)dx = (e^xcos(x) + e^xsin(x) + C + D)/(2)