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Integration by parts review

Review your integration by parts skills.

What is integration by parts?

Integration by parts is a method to find integrals of products:
u(x)v(x)dx=u(x)v(x)u(x)v(x)dx
or more compactly:
u dv=uvv du
We can use this method, which can be considered as the "reverse product rule," by considering one of the two factors as the derivative of another function.
Want to learn more about integration by parts? Check out this video.

Practice set 1: Integration by parts of indefinite integrals

Let's find, for example, the indefinite integral xcosxdx. To do that, we let u=x and dv=cos(x)dx:
xcos(x)dx=udv
u=x means that du=dx.
dv=cos(x)dx means that v=sin(x).
Now we integrate by parts!
xcos(x)dx=udv=uvvdu=xsin(x)sin(x)dx=xsin(x)+cos(x)+C
Remember you can always check your work by differentiating your result!
Problem 1.1
xe5xdx=?
Choose 1 answer:

Want to try more problems like this? Check out this exercise.

Practice set 2: Integration by parts of definite integrals

Let's find, for example, the definite integral 05xexdx. To do that, we let u=x and dv=exdx:
u=x means that du=dx.
dv=exdx means that v=ex.
Now we integrate by parts:
=05xexdx=05udv=[uv]0505vdu=[xex]0505exdx=[xexex]05=[ex(x+1)]05=e5(6)+e0(1)=6e5+1
Problem 2.1
1ex3lnx dx=?
Choose 1 answer:

Want to try more problems like this? Check out this exercise.

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