Integration by Parts

Integration by parts is a technique of integration applicable to integrands consisting of a product that cannot be rewritten as one or more easily integrated terms — at least, not without difficulty. The technique is particularly useful in cases containing a product of algebraic and transcendental factors.

Contents

Given two differentiable functions u and v,

$\int{u\,\,dv\,\,=\,\,uv\,\,-\,\,\int{v\,\,du}}$

To use the technique, one identifies suitable functions u and dv and then differentiates u to get du and integrate dv to get v — ignoring the usual constant of integration term, since it does not affect the final answer. Note that the rule can also be written as

$\int{u\,\,\frac{dv}{dx}\,\,dx}\,\,=\,\,uv\,\,-\,\,\int{\frac{du}{dx}\,\,v\,\,dx}$

There is a mnemonic for choosing u and dv, which covers a large variety of integrands:

u ⟸ L I A T E ⟹ dv

The letters stand for:

Logarithmic function (e.g., ln(x) and log_b(x))
Inverse trigonometric function (e.g., arctan(x), arcsin(x))
Algebraic function (e.g., x² and 3x)
Trigonometric function (e.g., sin(x) and cos(x))
Exponential function (e.g., e^x and 2^x)

This mnemonic only works when the integrand is the product of two different types of factors. The factor whose type of function appears higher in this list should generally be chosen as u, the factor whose type appears lower as dv.

Choosing u and dv

Integration by Parts boils down to selecting a factor, preferably the most complex, of the integrand that you can integrate either by direct integration or by the Substitution Method (also called u-substitution).

Now, think of all of the “traditional” functions you have encountered up to this point in mathematics, and ask yourself which ones you can easily integrate.

Basic Function Type	Easily Integrable?
Logarithmic (e.g., ln(x) and log_b(x))	No
Inverse Trigonometric (e.g., tan^-1(x) and csc^-1(x)	No
Algebraic (e.g., x³, √x, and 1/x)	Yes
Trigonometric (e.g., sin(x) and sec²(x), but not sec(x))	Yes (in some cases)
Exponential (e.g., e^x and 2^x)	Yes

Since we do not have integration formulas that allow us to integrate simple Logarithmic functions and Inverse trigonometric functions, it makes sense that they should not be chosen for . On the other hand, Exponential and Trigonometric functions are, in general, easy to integrate and make good choices for . Finally, we can always integrate basic Algebraic functions. However, their antiderivatives require slightly more work than the straightforward antiderivatives of the exponential and trigonometric functions. Therefore, if given the option, we would rather integrate exponential or trigonometric functions than algebraic functions.

Caution
Simpson sticks with finding dv first instead of finding u so that you get a more natural and understandable approach to Integration by Parts. All of the examples in chapter 2.3: Integration by Parts will reflect this.

Choosing dv

Given an integral where you wish to use Integration by Parts, the order of preference for choosing dv is as follows:

Exponential functions
Trigonometric functions
Algebraic functions
Inverse trigonometric functions
Logarithmic functions

We reverse the direction of our tactic for choosing dv and, instead, create the commonly-taught tactic for choosing u. If logarithms are the worst choice for dv, then they will be our first choice for u, and so on. This gives us the following tactic for choosing .

Choosing u

Given an integral where you wish to use Integration by Parts, the order of preference for choosing u is as follows:

Logarithmic functions
Inverse trigonometric functions
Algebraic functions
Trigonometric functions
Exponential functions

This last tactic gives a common mnemonic, LIATE, to take some of the guesswork out of our choices for u. The type of function in the integral that appears first in the list should be our first choice of u. When we have chosen u, dv is selected to be the remaining part of the integrand.

Note that we put LI at the beginning of the mnemonic; however, we could just as easily have started with IL, since these two types of functions won’t appear together in an Integration by Parts problem and they are both terrible choices for dv.

When to Use Integration By Parts

Use integration by parts for the integral of a product of two functions, specifically when u dv is easier to solve than the original integral. It is primarily used when functions are mixed (e.g., algebraic and trigonometric) and one becomes simpler when differentiated while the other is easy to integrate.

Products of different function types: When the integrand consists of a product of two essentially different functions (like algebraic and trigonometric, or algebraic and exponential) that are not related by a derivative pair (which would allow for standard u-substitution).
Stand-alone logarithmic or inverse functions: It is used to integrate functions like inverse trigonometric functions and natural logarithm (ln).

Understanding Integration by Parts Graphically

Let’s use the image below.

The area under the green curve (from the u-axis to the the curve) is defined by the expression in red. Thus we are adding the heights from the u-axis to the green curve (v) and multiplying by small changes in u (du).
The area under (to the left of) the green curve (from the v-axis to the the curve) is defined by the expression in blue. Thus we are adding the heights from the v-axis to the green curve (u) and multiplying by small changes in v (dv).
Adding these to areas together is the same as the are of the rectangle (u × v) bounded by the u and v-axis and the red and blue lines. We can express this by the following equation.

$\int{u\,\,dv\,\,+\,\,\int{v\,\,du\,\,=\,\,uv}}$

Subtracting ∫ v du from both sides provides us with the equation we are looking for.

$\int{u\,\,dv\,\,=\,\,uv\,\,-\,\,\int{v\,\,du}}$

References

apnorton. “When to do u-substitution and when to integrate by parts.” Mathematics Stack Exchange, October 24, 2013. https://math.stackexchange.com/q/538663.

Green, Larry. “2.4: Integration by Parts.” LibreTexts, November 7, 2013. https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Integral_Calculus/2%3A_Techniques_of_Integration/2.4%3A_Integration_by_Parts.

Simpson, Roy. “2.3: Integration by Parts.” LibreTexts, May 14, 2023. https://math.libretexts.org/Courses/Cosumnes_River_College/Math_401%3A_Calculus_II_-_Integral_Calculus/02%3A_Techniques_of_Integration/2.03%3A_Integration_by_Parts.

Additional Reading

“Integration by Parts.” Learning Lab – RMIT University, October 16, 2023. https://learninglab.rmit.edu.au/maths-statistics/integration/in8-integration-parts/.

“Integration by Parts.” Wikiversity. December 24, 2025. https://en.wikiversity.org/wiki/Integration_by_parts.

Kumar, Dilip. “Calculus Integration Review.” Medium, August 30, 2025. https://dilipkumar.medium.com/calculus-integration-review-8b1bd670d1c8.

“Master Integration by Parts: Solve Complex Integrals Easily.” StudyPug. Accessed July 13, 2026. https://www.studypug.com/calculus-help/integration-by-parts-2/?view=read.

[ ℰ ] McLean, Rachel. “Integration by Parts Explained.” Outlier, October 29, 2021. https://articles.outlier.org/understanding-integration-by-parts-in-calculus.

Pierce, Rod. “Integration by Parts.” Math Is Fun. Accessed July 13, 2026. https://www.mathsisfun.com/calculus/integration-by-parts.html.

Yashrajvishwakarma. “Calculus: Explained (pt.7) (Integration by parts, u-substitution).” Medium, November 22, 2022. https://medium.com/@yashrajvishwakarma.31/calculus-explained-pt-7-integration-by-parts-u-substitution-948be5cf5057.

Videos

Calculus 2: Integration by Parts (Video #1) | Math with Professor V

Introduction to integration by parts. Four examples demonstrating how to evaluate definite and indefinite integrals using integration by parts: includes boomerang problems, and using substitution then by parts.

How to Remember Integration by Parts (Without Memorizing the Formula)

Do you keep forgetting the integration by parts formula? In this video, I’ll show you the simple trick my high school calculus teacher taught me so I never have to memorize the formula. Once you understand where it comes from, you can rebuild it anytime you need it. We’ll work through several examples step-by-step and talk about how to choose u and dv so integration by parts becomes much easier.

The Easiest Way to Choose u and dv in Integration by Parts

How do you choose u and dv in integration by parts?

Should you use LIATE? LIPET? ALPES? Another acronym?

In this video, I’ll show you the simple principle that all of these acronyms are trying to teach. Once you understand the purpose of integration by parts, choosing u and dv becomes much more intuitive—and you won’t have to rely on memorizing a list of letters.

We’ll work through several examples step-by-step and discuss how to think about integration by parts like a mathematician instead of just following a mnemonic.

Notes

When integrating, functions that are “not related by a derivative pair” lack a direct parent-child relationship (such as f(x) and f'(x)). This means they cannot be evaluated by directly undoing a known derivative or through simple u-substitution.

In calculus, functions function in derivative pairs (e.g., sin(x) and cos (x), or x² and 2x). When solving an integral with such a pair—for example, ∫ cos(x) dx = sin(x) + C—you are reversing the derivative to find the original function.

When functions are not related by a derivative pair, it means one function inside the integral is not the exact derivative of the other. The standard reverse-differentiation formulas will not work directly. Instead, you must use more advanced techniques to “break apart” or simplify the expression so it can be integrated.

When to Do U-Substitution and When to Integrate by Parts

Always do a u-sub if you can; if you cannot, consider integration by parts.

A u-sub can be done whenever you have something containing a function (we’ll call this g), and that something is multiplied by the derivative of g. That is, if you have ∫ f(g(x)) g′(x) dx, use a u-sub.

Integration by parts is whenever you have two functions multiplied together–one that you can integrate, one that you can differentiate.

My strategy is to try to “play it out” in my mind and try to see which one will work better. The best way to get better at these sorts of integrals is to practice large sets of each type. Then, you start to think “Oh–this looks like a u-sub!” or, “maybe by-parts is better for this.” Practice is really the best way to get better at recognizing each type. [apnorton]

Boomerang

“Boomerang” (or loop) problems in integration by parts occur when applying the method twice yields a new integral that is a constant multiple of your original integral. Instead of looping infinitely, you move the identical integral to the left side and solve algebraically.

Integral of the Day 6.7.25 | Classic BOOMERANG Problem–Can You Solve It? | Math with Professor V

When do you use this approach?

Watch out for exponential and sine/cosine combinations where neither function reduces to zero when differentiated. While a standard integral like ∫ x cos(x) dx finishes in one step, a function like ∫ e^ax cos(bx) dx will always require the boomerang method.

For practice, you can look at the Calculus II Integration by Parts Practice Problems provided by Paul’s Online Math Notes.

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The featured image on this page is from the Integral Wallpaper website.