Knowunity AI

More

Careers

Creator Program

Open the App

Subjects

AP Calculus AB/BCAP Calculus AB/BC26 views·Updated May 20, 2026·2 pages

Integration by Parts in Calculus BC

user profile picture
Michelle@michylol

Integration by parts is a powerful technique for solving complex... Show more

1
of 2
9/15173 Integration by Parts $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$ $d(uv) = udv + vdu$ $d(uv) - vdu = udv$ $udv = d(uv

Integration by Parts Formula

Integration by parts uses the formula ∫u dv = uv - ∫v du, which comes from rearranging the product rule for derivatives. This technique is especially useful when you have a product of functions and direct integration seems impossible.

When choosing which function to call "u," remember the helpful acronym "LIATE" which stands for Logarithmic, Inverse trigonometric, Algebraic (polynomials), Trigonometric, and Exponential functions. Functions earlier in this list make better choices for "u."

For example, to solve ∫3x+13x+1(sin 5x)dx, we set u = 3x+1 and dv = sin 5x dx. After finding du = 3dx and v = -⅕cos 5x, we substitute into our formula: uv - ∫v du. This gives us -⅕3x+13x+1(cos 5x) + ⅗∫cos 5x dx, which simplifies to -⅕3x+13x+1(cos 5x) + ⅗(⅕sin 5x) + c.

💡 Pro Tip: When choosing which function to be "u" and which to be "dv," pick the function for "u" that simplifies when differentiated, and the function for "dv" that remains manageable when integrated.

2
of 2
9/15173 Integration by Parts $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$ $d(uv) = udv + vdu$ $d(uv) - vdu = udv$ $udv = d(uv

Tabular Integration Method

The tabular method is a shortcut for integration by parts when you need to apply the technique multiple times. It's especially useful for integrals involving products of polynomials with trigonometric or exponential functions.

To use this method, arrange the functions in a table with derivatives of one function down the left side and integrals of the other function down the right. Continue until the left column reaches zero. Then multiply diagonally, alternating plus and minus signs, to get your answer.

For example, with ∫x³sin x dx, we put x³ on the left (since it differentiates to zero eventually) and sin x on the right. We differentiate x³ repeatedly until we reach 0, and integrate sin x repeatedly. The final answer is -x³cos x + 3x²sin x - 6x cos x - 6sin x + C.

🔑 Remember: The tabular method saves time when integrating products where one function differentiates to zero after several steps (like polynomials) and the other has simple repeated integrals likesin/cosorexponentialslike sin/cos or exponentials.

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.

Most popular content in AP Calculus AB/BC

9

Most popular content

9

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

Students love us — and so will you.

4.6/5App Store
4.7/5Google 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 SiOS 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 KlichAndroid 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.

AnnaiOS user

AP Calculus AB/BCAP Calculus AB/BC26 views·Updated May 20, 2026·2 pages

Integration by Parts in Calculus BC

user profile picture
Michelle@michylol

Integration by parts is a powerful technique for solving complex integrals when simple substitution won't work. It's based on the product rule for derivatives but used in reverse, allowing you to transform difficult integrals into simpler ones that you can... Show more

1
of 2
9/15173 Integration by Parts $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$ $d(uv) = udv + vdu$ $d(uv) - vdu = udv$ $udv = d(uv

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

Integration by Parts Formula

Integration by parts uses the formula ∫u dv = uv - ∫v du, which comes from rearranging the product rule for derivatives. This technique is especially useful when you have a product of functions and direct integration seems impossible.

When choosing which function to call "u," remember the helpful acronym "LIATE" which stands for Logarithmic, Inverse trigonometric, Algebraic (polynomials), Trigonometric, and Exponential functions. Functions earlier in this list make better choices for "u."

For example, to solve ∫3x+13x+1(sin 5x)dx, we set u = 3x+1 and dv = sin 5x dx. After finding du = 3dx and v = -⅕cos 5x, we substitute into our formula: uv - ∫v du. This gives us -⅕3x+13x+1(cos 5x) + ⅗∫cos 5x dx, which simplifies to -⅕3x+13x+1(cos 5x) + ⅗(⅕sin 5x) + c.

💡 Pro Tip: When choosing which function to be "u" and which to be "dv," pick the function for "u" that simplifies when differentiated, and the function for "dv" that remains manageable when integrated.

2
of 2
9/15173 Integration by Parts $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$ $d(uv) = udv + vdu$ $d(uv) - vdu = udv$ $udv = d(uv

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

Tabular Integration Method

The tabular method is a shortcut for integration by parts when you need to apply the technique multiple times. It's especially useful for integrals involving products of polynomials with trigonometric or exponential functions.

To use this method, arrange the functions in a table with derivatives of one function down the left side and integrals of the other function down the right. Continue until the left column reaches zero. Then multiply diagonally, alternating plus and minus signs, to get your answer.

For example, with ∫x³sin x dx, we put x³ on the left (since it differentiates to zero eventually) and sin x on the right. We differentiate x³ repeatedly until we reach 0, and integrate sin x repeatedly. The final answer is -x³cos x + 3x²sin x - 6x cos x - 6sin x + C.

🔑 Remember: The tabular method saves time when integrating products where one function differentiates to zero after several steps (like polynomials) and the other has simple repeated integrals likesin/cosorexponentialslike sin/cos or exponentials.

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.

Most popular content in AP Calculus AB/BC

9

Most popular content

9

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

Students love us — and so will you.

4.6/5App Store
4.7/5Google 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 SiOS 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 KlichAndroid 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.

AnnaiOS user