•

Feb 19, 2026

•

3 pages

Understanding Integration Using U-Substitution

Chelsea

@zuncho

Integration by substitution, or u-substitution, is a powerful technique that... Show more

1 / 3

$12 Section 4.8: Integration by substitution (u-substitution) Example Find $ \int (x^2 + 1)^2 (2x) dx $ Method I $ \int (x^2+1)^2 (2x)dx $ $$

U-Substitution Basics

U-substitution works like a puzzle piece that fits perfectly into complex integration problems. When you see expressions like $(x^2 + 1)^2(2x)$ , you can often simplify the work dramatically by making a smart substitution.

When choosing your substitution, look for expressions inside parentheses to use as your "u" value. Then find something in the integral that resembles the derivative of u to use as "du". For example, if $u = x^2 + 1$ , then $du = 2x,dx$ , which means you can replace $2x,dx$ in your integral.

💡 Pro Tip: Instead of expanding complex expressions and integrating term by term (the long way), save time by identifying substitution patterns. Look for expressions where one part resembles the derivative of another.

Let's see u-substitution in action with $\int (x^2+1)^2 (2x) dx$ . If we set $u = x^2 + 1$ , then $du = 2x,dx$ . The integral transforms to $\int u^2 du = \frac{u^3}{3} + C = \frac{(x^2+1)^3}{3} + C$ . That's much simpler than expanding the expression first!

$12 Section 4.8: Integration by substitution (u-substitution) Example Find $ \int (x^2 + 1)^2 (2x) dx $ Method I $ \int (x^2+1)^2 (2x)dx $ $$

Applying U-Substitution

U-substitution works for many types of integrals. The key is identifying patterns where one part of the integrand resembles the derivative of another part. Let's explore some common patterns:

For rational functions like $\int \frac{-4x}{(1-2x^2)^2} dx$ , set $u = 1-2x^2$ so $du = -4x,dx$ . This transforms the integral to $\int \frac{1}{u^2} du = -\frac{1}{u} + C = -\frac{1}{1-2x^2} + C$ .

Trigonometric functions also work well with u-substitution. For $\int 5\cos(5x)dx$ , let $u = 5x$ , then $du = 5,dx$ . This gives us $\int \cos(u) \frac{du}{5} = \frac{1}{5}\sin(u) + C = \sin(5x) + C$ .

⚠️ Important: You can only multiply an integral by a constant, not a variable! Don't try to force a substitution by introducing variable factors outside the integral.

Exponential and logarithmic functions follow the same principles. For $\int 2x^3e^{x^4}dx$ , let $u = x^4$ , then $du = 4x^3dx$ . Since we have $2x^3 $instead of$ 4x^3 $, we adjust:$ \int 2x^3e^{x^4}dx = \frac{1}{2}\int e^u du = \frac{1}{2}e^{x^4} + C$.

$12 Section 4.8: Integration by substitution (u-substitution) Example Find $ \int (x^2 + 1)^2 (2x) dx $ Method I $ \int (x^2+1)^2 (2x)dx $ $$

Definite Integrals with U-Substitution

When using u-substitution with definite integrals, remember to change the limits of integration to match your new variable. This avoids having to substitute back at the end!

For example, with $\int_{0}^{1} x\sqrt{x^2 + 1}dx$ , we set $u = x^2 + 1$ and $du = 2x,dx$ . When $x = 0$ , $u = 1$ , and when $x = 1$ , $u = 2$ . The integral becomes $\frac{1}{2}\int_{1}^{2} u^{1/2}du = \frac{1}{3}[(x^2 + 1)^{3/2}]_{0}^{1} = \frac{1}{3}[\sqrt{8} - 1]$ .

More complex substitutions require careful tracking of your variables. For $\int_{0}^{5} \frac{2x + 1}{\sqrt{x+4}}dx$ , set $u = x + 4$ . This means $x = u - 4$ and $dx = du$ . We also need to adjust the limits: when $x = 0$ , $u = 4$ , and when $x = 5$ , $u = 9$ .

🌟 Visualization Tip: Think of u-substitution as "zooming in" on the complicated part of the integral. By focusing on just that part, the whole problem becomes clearer.

The algebraic manipulation can get tricky, especially when substituting expressions with multiple terms. Take your time to rewrite the integrand carefully in terms of u before proceeding with the integration.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5

App Store

4.7/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

AP Calculus AB/BC

•

Feb 19, 2026

•

3 pages

Understanding Integration Using U-Substitution

Chelsea

@zuncho

Integration by substitution, or u-substitution, is a powerful technique that simplifies complex integrals by changing variables. This method transforms difficult integrals into more manageable forms by replacing part of the expression with a new variable u, making integration much easier... Show more

$12 Section 4.8: Integration by substitution (u-substitution) Example Find $ \int (x^2 + 1)^2 (2x) dx $ Method I $ \int (x^2+1)^2 (2x)dx $ $$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

U-Substitution Basics

💡 Pro Tip: Instead of expanding complex expressions and integrating term by term (the long way), save time by identifying substitution patterns. Look for expressions where one part resembles the derivative of another.

$12 Section 4.8: Integration by substitution (u-substitution) Example Find $ \int (x^2 + 1)^2 (2x) dx $ Method I $ \int (x^2+1)^2 (2x)dx $ $$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Applying U-Substitution

U-substitution works for many types of integrals. The key is identifying patterns where one part of the integrand resembles the derivative of another part. Let's explore some common patterns:

For rational functions like $\int \frac{-4x}{(1-2x^2)^2} dx$ , set $u = 1-2x^2$ so $du = -4x,dx$ . This transforms the integral to $\int \frac{1}{u^2} du = -\frac{1}{u} + C = -\frac{1}{1-2x^2} + C$ .

⚠️ Important: You can only multiply an integral by a constant, not a variable! Don't try to force a substitution by introducing variable factors outside the integral.

$12 Section 4.8: Integration by substitution (u-substitution) Example Find $ \int (x^2 + 1)^2 (2x) dx $ Method I $ \int (x^2+1)^2 (2x)dx $ $$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Definite Integrals with U-Substitution

When using u-substitution with definite integrals, remember to change the limits of integration to match your new variable. This avoids having to substitute back at the end!

🌟 Visualization Tip: Think of u-substitution as "zooming in" on the complicated part of the integral. By focusing on just that part, the whole problem becomes clearer.

We thought you’d never ask...

What is the Knowunity AI companion?

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Smart Tools NEW

Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Practice Test ✓ Essay Outlines

Practice Test

Quiz

Flashcards

Essay

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5

App Store

4.7/5

Google Play

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Understanding Integration Using U-Substitution

U-Substitution Basics

Applying U-Substitution

Definite Integrals with U-Substitution

We thought you’d never ask...

What is the Knowunity AI companion?

Where can I download the Knowunity app?

Is Knowunity really free of charge?

Most popular content: U-substitution

AP Calculus AB | Integration using U-Substitution (Indefinite Integrals)

U-Substitution

Most popular content in AP Calculus AB/BC

Riemann Sums & Properties of the Definite Integral

Continuity & The Intermediate Value Theorem

AP Calculus AB | Integration using U-Substitution (Indefinite Integrals)

Rules of Differentiation

AP Calculus AB

AP Calc 2.2 Notes

Derivative and Integral Review

U-Substitution

Limits and Continuity

Most popular content

FAR NOTES- CPM

AFAR NOTES- CPM

TAX NOTES - CPM

FAR NOTES-HERCULES

AFAR NOTES- HERCULES

MS NOTES-CPM

RFBT NOTES- HERCULES

FAR NOTES - KUYA WOWOWIE

RFBT NOTES- CPM

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5

4.7/5

Understanding Integration Using U-Substitution

Sign up to see the contentIt's free!

U-Substitution Basics

Sign up to see the contentIt's free!

Applying U-Substitution

Sign up to see the contentIt's free!

Definite Integrals with U-Substitution

We thought you’d never ask...

What is the Knowunity AI companion?

Where can I download the Knowunity app?

Is Knowunity really free of charge?

Most popular content: U-substitution

AP Calculus AB | Integration using U-Substitution (Indefinite Integrals)

U-Substitution

Most popular content in AP Calculus AB/BC

Riemann Sums & Properties of the Definite Integral

Continuity & The Intermediate Value Theorem

AP Calculus AB | Integration using U-Substitution (Indefinite Integrals)

Rules of Differentiation

AP Calculus AB

AP Calc 2.2 Notes

Derivative and Integral Review

U-Substitution

Limits and Continuity

Most popular content

FAR NOTES- CPM

AFAR NOTES- CPM

TAX NOTES - CPM

FAR NOTES-HERCULES

AFAR NOTES- HERCULES

MS NOTES-CPM

RFBT NOTES- HERCULES

FAR NOTES - KUYA WOWOWIE

RFBT NOTES- CPM

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5

4.7/5