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Mathematics

Integration by Parts

LIATE, the Tabular Method, and Cyclic Integrals — A TLDR Primer

Integration by parts shows up on almost every calculus exam — and it's the technique that trips students up the most. The formula looks simple enough, but knowing when to use it, how to pick the right pieces, and what to do when it loops back on itself is where students lose points.

This TLDR guide covers everything you need to handle integration by parts with confidence. You'll see exactly where the formula comes from (it's just the product rule in reverse), how to apply the LIATE rule so choosing *u* and *dv* becomes routine, and how to work through the one-pass examples that appear most often on tests. From there, the guide walks through repeated application and the tabular method — the shortcut that turns long polynomial-times-exponential problems into a clean grid. A dedicated section handles cyclic integrals like $e^x \sin x$, where integration by parts seems to go in circles until you solve for the answer algebraically. The final section applies everything to definite integrals and calls out the mistakes students make most often, so you don't lose easy points to boundary-term errors.

This is a calculus 2 help resource for college students as well as a focused ap calculus bc exam prep companion — short enough to read in one sitting, detailed enough to replace a full chapter of lecture notes. No filler, no padding: just the formula, the method, and enough worked examples to make it stick.

Grab it before your next exam.

What you'll learn
  • State the integration by parts formula and explain where it comes from.
  • Use the LIATE rule to choose u and dv reliably.
  • Apply integration by parts repeatedly, including tabular integration.
  • Handle cyclic cases where the original integral reappears.
  • Evaluate definite integrals using integration by parts.
What's inside
  1. 1. Where the Formula Comes From
    Derives the integration by parts formula from the product rule and shows what kinds of integrals it is built for.
  2. 2. Choosing u and dv: The LIATE Rule
    Teaches the standard heuristic for picking u and dv so the new integral is simpler than the original.
  3. 3. Worked Examples: One Pass
    Walks through the canonical one-application examples step by step, including products of polynomials with exponentials, sines, and logarithms.
  4. 4. Repeated Integration by Parts and the Tabular Method
    Shows how to apply the formula multiple times and introduces the tabular shortcut for polynomial-times-easy-function integrals.
  5. 5. Cyclic Integrals and Solving for the Answer
    Handles cases like e^x sin x where integration by parts loops back to the original integral, and you solve algebraically.
  6. 6. Definite Integrals and Common Pitfalls
    Applies integration by parts to definite integrals, evaluates the boundary term carefully, and lists the mistakes students make most often.
Published by Solid State Press
Integration by Parts cover
TLDR STUDY GUIDES

Integration by Parts

LIATE, the Tabular Method, and Cyclic Integrals — A TLDR Primer
Solid State Press

Integration by Parts

LIATE, the Tabular Method, and Cyclic Integrals — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 Where the Formula Comes From
  2. 2 Choosing u and dv: The LIATE Rule
  3. 3 Worked Examples: One Pass
  4. 4 Repeated Integration by Parts and the Tabular Method
  5. 5 Cyclic Integrals and Solving for the Answer
  6. 6 Definite Integrals and Common Pitfalls
Chapter 1

Where the Formula Comes From

Every differentiation rule has a mirror image in integration. The product rule is the mirror you need here.

Recall the product rule: if $u$ and $v$ are both differentiable functions of $x$, then

$\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}.$

Now integrate both sides with respect to $x$:

$\int \frac{d}{dx}[uv]\, dx = \int u\frac{dv}{dx}\, dx + \int v\frac{du}{dx}\, dx.$

The left side is just $uv$ — integrating a derivative gives you back the original function. Rearranging:

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

Writing $\frac{dv}{dx}\,dx$ as $dv$ and $\frac{du}{dx}\,dx$ as $du$, this becomes the integration by parts formula:

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

That is the whole derivation. No magic — just the product rule, integrated.

What the Formula Actually Does

The formula trades one integral for another. You start with $\int u\, dv$, which you cannot easily evaluate, and you end with $\int v\, du$, which you hope is simpler. The word "hope" is doing real work in that sentence: the formula only helps when the new integral is genuinely easier than the original. Choosing $u$ and $dv$ badly can produce something worse. Choosing well is the main skill, and the next subsection gives you a reliable method for doing that.

For now, notice what each piece requires of you. You need to split the integrand into two factors: one you call $u$ and one you call $dv$. From $u$ you compute $du$ by differentiating. From $dv$ you compute $v$ by integrating. Then you assemble $uv - \int v\, du$.

A common mistake is to forget that $dv$ must include $dx$. If your integrand is $x e^x\, dx$, then $dv$ might be $e^x\, dx$ — the $dx$ belongs with $dv$, not off to the side.

What Kinds of Integrals Need This

About This Book

If you are a high school student working through AP Calculus BC exam prep, a college freshman grinding through Calculus 2, or anyone who has stared at an integral involving a product of functions and had no idea where to start, this book was written for you. It also works for tutors who need a fast, reliable review before a session.

This integration by parts calculus study guide covers the formula and where it comes from, how to choose u and dv using the LIATE rule, and how to do integration by parts step by step through a sequence of worked examples. It also covers the tabular method for repeated applications, cyclic integrals, and definite integrals with boundary terms. A concise overview with no filler.

Read straight through once, then work every example yourself before reading the solution. Finish with the problem set at the end. Treat this as a calculus study guide for high school students and early college students serious about mastering one of the most useful integration techniques in the course.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon