Calculus:
Integral: Integration by Parts

1.  
∫ln⁡xx5 dx\int \frac{\ln{x}}{{x}^{5}} \, dx
  Solution
2.  
∫exsin⁡x dx\int {e}^{x}\sin{x} \, dx
  Solution
3.  
∫xcos⁡3x dx\int x\cos{3x} \, dx
  Solution
4.  
∫sin⁡−1(y) dy\int \sin^{-1}{(y)} \, dy
  Solution
5.  
∫cos⁡xx2 dx\int \cos{x}{x}^{2} \, dx
  Solution
6.  
∫ln⁡x dx\int \ln{x} \, dx
  Solution
7.  
∫tan⁡2xsec⁡x dx\int \tan^{2}x\sec{x} \, dx
  Solution
8.  
∫cos⁡−1(y) dy\int \cos^{-1}{(y)} \, dy
  Solution
9.  
∫xex dx\int x{e}^{x} \, dx
  Solution
10.  
∫xln⁡x dx\int x\ln{x} \, dx
  Solution

Integration by Parts - Introduction

Integration by parts is another important technique of integration. You might ask, when should I use it? The general rule is, whenever you have a product of two functions, one that you can integrate, and one that you can differentiate. In the context of other integration techniques, generally speaking, you should always use integration by substitution first if you can. If you cannot, then consider integration by parts.
The formula for integration by parts is:
∫u dv=uv−∫v du\int u \, dv=uv-\int v \, du
The critical step is to correctly identify
uu
and
dvdv
. Let’s look at an example in the next section.

An Example of Integration by Parts

Let's find the integral of the following function:
∫xex dx\int x{e}^{x} \, dx
If you were to try integration by substitution first (which you should), you would have seen that it did not work. Now, let us try integration by parts. The dilemma arises: what should we pick as
uu
and
dvdv
in the following formula?
∫u dv=uv−∫v du\int u \, dv=uv-\int v \, du
One thing for sure is, we want to pick
dvdv
to be something that we can integrate.
ex{e}^{x}
would be a good candidate, since the integral of
ex{e}^{x}
is itself. Let's try the following:
u=xu=x
,
dv=exdv={e}^{x}
,
du=dxdu=dx
and
v=exv={e}^{x}
. This gives:
xex−∫ex dxx{e}^{x}-\int {e}^{x} \, dx
This looks good, since we are now left with integrating
ex{e}^{x}
, which we know how to do. It is simply
ex{e}^{x}
. Finally, do not forget to add the constant
CC
. This produces the final answer:
xex−ex+Cx{e}^{x}-{e}^{x}+C
We are now done. Note that if you picked the wrong
uu
and
dvdv
on the first try, don't worry. You would have seen that it did not work out, and you would just have to pick another pair. With enough practice, you will be able to make the right choices within the first couple tries.

What's Next

Best bet? Start with the practice problems at the top of this page to get more familiar with integration by parts. Want to dig deeper? Sign up for Cymath Plus today! You can also check out the Cymath app for iOS and Android.
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