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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 6

Lesson 13: Using integration by parts

# Integration by parts review

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.E (LO)
,
FUN‑6.E.1 (EK)
Review your integration by parts skills.

## What is integration by parts?

Integration by parts is a method to find integrals of products:
integral, u, left parenthesis, x, right parenthesis, v, prime, left parenthesis, x, right parenthesis, d, x, equals, u, left parenthesis, x, right parenthesis, v, left parenthesis, x, right parenthesis, minus, integral, u, prime, left parenthesis, x, right parenthesis, v, left parenthesis, x, right parenthesis, d, x
or more compactly:
integral, u, space, d, v, equals, u, v, minus, integral, v, space, d, u
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.

## Practice set 1: Integration by parts of indefinite integrals

Let's find, for example, the indefinite integral integral, x, cosine, x, d, x. To do that, we let u, equals, x and d, v, equals, cosine, left parenthesis, x, right parenthesis, d, x:
integral, x, cosine, left parenthesis, x, right parenthesis, d, x, equals, integral, u, d, v
u, equals, x means that d, u, equals, d, x.
d, v, equals, cosine, left parenthesis, x, right parenthesis, d, x means that v, equals, sine, left parenthesis, x, right parenthesis.
Now we integrate by parts!
\begin{aligned} \displaystyle\int x\cos(x)\,dx &=\displaystyle\int u\,dv \\\\ &=uv-\displaystyle\int v\,du \\\\ &=\displaystyle x\sin(x)-\int\sin(x)\,dx \\\\ &=x\sin(x)+\cos(x)+C \end{aligned}
Problem 1.1
• Current
integral, x, e, start superscript, 5, x, end superscript, d, x, equals, question mark

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 integral, start subscript, 0, end subscript, start superscript, 5, end superscript, x, e, start superscript, minus, x, end superscript, d, x. To do that, we let u, equals, x and d, v, equals, e, start superscript, minus, x, end superscript, d, x:
u, equals, x means that d, u, equals, d, x.
d, v, equals, e, start superscript, minus, x, end superscript, d, x means that v, equals, minus, e, start superscript, minus, x, end superscript.
Now we integrate by parts:
\begin{aligned} &\phantom{=}\displaystyle\int_0^5 xe^{-x}\,dx \\\\ &=\displaystyle\int_0^5 u\,dv \\\\ &=\Big[uv\Big]_0^5-\displaystyle\int_0^5 v\,du \\\\ &=\displaystyle\Big[ -xe^{-x}\Big]_0^5-\int_0^5-e^{-x}\,dx \\\\ &=\Big[-xe^{-x}-e^{-x}\Big]_0^5 \\\\ &=\Big[-e^{-x}(x+1)\Big]_0^5 \\\\ &=-e^{-5}(6)+e^0(1) \\\\ &=-6e^{-5}+1 \end{aligned}
Problem 2.1
• Current
integral, start subscript, 1, end subscript, start superscript, e, end superscript, x, cubed, natural log, x, space, d, x, equals, question mark

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

## Want to join the conversation?

• in the int (0 -> pi) of xsin(2x)dx problem, in the solution, the third to last line, shouldn't that be (sin(2x)/4) not (sin(4x)/4)? or am I missing something?
• You are correct, it is a typo, though it does not effect the result since sin(nπ) = 0 for all integers n.
You can, and should, report the error since you found it.
• Like Bhoovesh I am also fuzzy about the compact notation. It seems that the confusion is not with Leibniz notation vs Newton's, but rather I am concerned about falling in a pit as a consequence of having only one letter in an expression for which I am accustomed to two. The dropping of the x's and dx's makes me nervous about getting fouled up as a consequence of x not being the only variable. Where is y, and how does one keep track of it with the more compact notation? I would like to know the conventions and rules for this.
• Why hasn't Sal explained about the compact form of Integration by parts??i don't understand it!! It contradicts to what Sal said about differentials earlier that the differentials are not numbers or function which can't cancelled or algebraically manipulated!!
• The "compact form" is just a different way to write the form used in the videos. Basically, the only difference is that the "video form" uses prime notation (f'(x)), and the "compact form" uses Leibniz notation (dy/dx). If you are used to the prime notation form for integration by parts, a good way to learn Leibniz form is to set up the problem in the prime form, then do the substitutions f(x) = u, g'(x)dx = dv, f'(x) = v, g(x)dx = du. At least, that's how it clicked for me.

As far as the manipulating differentials goes, it's true that you can't just treat differentials like they are normal terms in an equation (as if dx were the variable d times the variable x), but it is legal to split up the dy/dx when differentiating both sides of an equation. The concept here is exactly the same as what is used when doing u-substitution (URL to video below if you need it).
Hope this helps, and good luck with your work!
• What is the use of integration? When do we use it in our practical lives?
(1 vote)
• we can use integration to find Displacement from Velocity, and Velocity from Acceleration, Voltage across a Capacitor, Moments of Inertia by Integration and many more...
• Why does the integral of e^5x dx = 1/5 e^5x? Is it an application of the reverse chain rule?
Thanks very much!
• That's one way of thinking about it, yes. As you continue on in math, this will become almost second-nature and you won't even think about the chain rule when integrating simple exponential functions.
• How would you integrate something in parentheses, like (x^2 +1)^1/2?
• That depends hugely on what's in the parentheses. √(x²+1) can be done by trigonometric substitution, but √(x³+1) cannot be done by elementary means.
• Is there a reverse division rule that can sometimes serve as a substitute for this? An example where this would be useful is (ln(5x))/(x^2)