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Dec 13, 2025
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Integration using U-Substitution is a powerful technique that helps solve... Show more
Ever looked at a complicated integral and felt stuck? U-substitution is your secret weapon! This technique works by cleverly replacing part of an integral with a simpler variable (u) to make the whole problem easier to solve.
When choosing what to substitute, look for specific patterns. For composite functions like sin(ln x), choose the inside expression (ln x). With powered functions like 4x², choose the base . For rational functions, the denominator usually makes the best substitution.
Following these five steps makes u-substitution straightforward: choose u, differentiate u, substitute into the original integral, integrate the simplified expression, and finally re-substitute to get your answer in terms of the original variable.
💡 Success Tip: When you see a function inside another function, that's your clue to try u-substitution. The "inside" function often makes the perfect u!
Let's see this in action with examples like ∫3/dy. By setting u = 5-4y and finding du = -4dy, we can transform our integral into a basic form that's easy to solve, giving us -¾ln|5-4y| + C as our answer.
U-substitution becomes second nature with practice! When facing integrals with roots like ∫6/√da, start by identifying the expression under the root sign. Setting u = 3a+4 transforms this into a standard integral format.
Complex trigonometric integrals might look intimidating, but they're perfect for u-substitution. With ∫3sin(x)/√dx, setting u = 16+9cos(x) allows you to reorganize the integral into a familiar form involving √u.
The substitution process follows a consistent pattern: identify what to substitute, find the derivative, adjust your integral with the new variable, solve the simplified version, and convert back to the original variable. This methodical approach works even for intimidating integrals with inverse trigonometric functions.
🔍 Understanding Check: When you make a substitution, you're not just replacing variables - you're transforming the entire integral. Make sure to account for dx (or whatever your variable of integration is) in terms of du!
With rational expressions like ∫12/dx, a substitution of u = 49+4x quickly converts this to a basic integral involving 1/u, which we know equals ln|u|.
Ready to tackle tougher problems? Integrals with exponential terms like ∫e^dt become approachable when you focus on the exponent. By setting u = 3t²+4t-1, you transform the integral into a form involving e^u.
Sometimes you'll need to factor expressions to identify the best substitution. When you do this correctly, complex problems simplify dramatically. For example, with ∫5g/dg, setting u = 6g²+7 converts this to a much simpler integral.
Trigonometric integrals are perfect candidates for u-substitution. When integrating ∫3sin(x)cos³(x)dx, substituting u = cos(x) transforms the integral into a power function that's straightforward to evaluate.
🌟 Pro Strategy: Look for patterns where one part of the expression resembles the derivative of another part. That's often your clue for the perfect substitution!
With exponentials involving polynomials, like ∫5x²e^(x³)dx, setting u = x³ transforms the problem into a basic exponential integral. This shows how u-substitution can turn seemingly complex problems into ones you already know how to solve.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
App Store
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 THE SCHOOLGPT. 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
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 THE SCHOOLGPT. 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
Integration using U-Substitution is a powerful technique that helps solve complex integrals by reversing the chain rule from calculus. This method transforms difficult integrals into simpler ones by making a strategic substitution, making seemingly complicated problems much more manageable.
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Improve your grades
Join milions of students
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Ever looked at a complicated integral and felt stuck? U-substitution is your secret weapon! This technique works by cleverly replacing part of an integral with a simpler variable (u) to make the whole problem easier to solve.
When choosing what to substitute, look for specific patterns. For composite functions like sin(ln x), choose the inside expression (ln x). With powered functions like 4x², choose the base . For rational functions, the denominator usually makes the best substitution.
Following these five steps makes u-substitution straightforward: choose u, differentiate u, substitute into the original integral, integrate the simplified expression, and finally re-substitute to get your answer in terms of the original variable.
💡 Success Tip: When you see a function inside another function, that's your clue to try u-substitution. The "inside" function often makes the perfect u!
Let's see this in action with examples like ∫3/dy. By setting u = 5-4y and finding du = -4dy, we can transform our integral into a basic form that's easy to solve, giving us -¾ln|5-4y| + C as our answer.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
U-substitution becomes second nature with practice! When facing integrals with roots like ∫6/√da, start by identifying the expression under the root sign. Setting u = 3a+4 transforms this into a standard integral format.
Complex trigonometric integrals might look intimidating, but they're perfect for u-substitution. With ∫3sin(x)/√dx, setting u = 16+9cos(x) allows you to reorganize the integral into a familiar form involving √u.
The substitution process follows a consistent pattern: identify what to substitute, find the derivative, adjust your integral with the new variable, solve the simplified version, and convert back to the original variable. This methodical approach works even for intimidating integrals with inverse trigonometric functions.
🔍 Understanding Check: When you make a substitution, you're not just replacing variables - you're transforming the entire integral. Make sure to account for dx (or whatever your variable of integration is) in terms of du!
With rational expressions like ∫12/dx, a substitution of u = 49+4x quickly converts this to a basic integral involving 1/u, which we know equals ln|u|.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Ready to tackle tougher problems? Integrals with exponential terms like ∫e^dt become approachable when you focus on the exponent. By setting u = 3t²+4t-1, you transform the integral into a form involving e^u.
Sometimes you'll need to factor expressions to identify the best substitution. When you do this correctly, complex problems simplify dramatically. For example, with ∫5g/dg, setting u = 6g²+7 converts this to a much simpler integral.
Trigonometric integrals are perfect candidates for u-substitution. When integrating ∫3sin(x)cos³(x)dx, substituting u = cos(x) transforms the integral into a power function that's straightforward to evaluate.
🌟 Pro Strategy: Look for patterns where one part of the expression resembles the derivative of another part. That's often your clue for the perfect substitution!
With exponentials involving polynomials, like ∫5x²e^(x³)dx, setting u = x³ transforms the problem into a basic exponential integral. This shows how u-substitution can turn seemingly complex problems into ones you already know how to solve.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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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 THE SCHOOLGPT. 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
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 THE SCHOOLGPT. 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
AP Calculus AB/BC
Algebra
Algebra 1
Overige