U-substitution with integration by parts (KristaKingMath)
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- Опубликовано: 10 фев 2025
- ► My Integrals course: www.kristaking...
Learn how to find the integral of a function using u-substitution and then integration by parts. Also, since this is a definite integral, evaluate at the limits of integration.
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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student-from basic middle school classes to advanced college calculus-figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: www.kristakingm...
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Thanks! You helped me out. And you are the prettiest math teacher ever.
here voice is so soft. makes math more soothing.
Girl you are awesome. My calc book had a problem like this but it never showed how to use the substitution. Now I see why it never showed how to do it, its super easy!
I'm glad I could help! :D
much better than the Stewart text book example
You're welcome! I'm so glad I can help. :)
Clear as can be. Hallmark of your production.
For all those who have a test or final tomorrow, Good luck!
Lmaooooo thankssss
I am learning now for a test that I have in 20 min
You're great :) I haven't found anyone else on youtube that explains mathematics so effective and simple :D
Thanks for the kind words :)
Thank you for the video. You made it very easy to follow along.
your videos always help out a lot thanks for all the effort that you put in for us
Although for this problem at wolframalpha, computational time was exceeded, I got the solution to another problem at wolframalpha. Thanks for the link:)
Thank you again. This was the perfect pace for me. Not to fast and not to slow. :-)
You're welcome!!
Glad you think so! :D
Thanks this was very helpful! You have beautiful hand writing!
incredible. .extraordinary concepts.Thanks for guidance
You're welcome! I'm glad it could help.
This is incredible! THANK YOU SO MUCH!
You're welcome, I'm so glad you liked it!
I LOVE YOU I FINALLY UNDERSTAND
:D
Why can't you be my teacher? You're amazing at explaining this stuff. I have followed you since pre-calc and I'm now in calc 2.
I'm honored! And even though I can't technically be your teacher, I can still be your teacher on RUclips!
I'm so glad it was helpful! :D
this is very helpful thank you so much
Thank you so much. I really appreciate your help. Extremely helpfull
You're welcome! :)
This was so helpful! You're amazing!
+Melinda Singh I'm so glad it helped!!
OMG!!! this helps so much!!!! thank you
it's just more accurate to write pi as a representation of it's decimal approximation than it is to round off the decimal value. :)
You made a mistake with the x-substitution concerning the limits cause they should be pi and pi/2..
Understand this so much better now. Thank you! Do you have a tutorial video of something like this using substitution with ln or inverse trig functions?
I'm so glad! :) These might help:
U-Substitution Example 8
U-Substitution Example 5
Integrals of Inverse Hyperbolic Functions
now i love math hahahah thanks soooooooooooo much !!!!
Very helpful, thank you!
So glad it could help!
Hi. Since the computational time was exceeded I solved the problem myself. Here it goes:
∫((x²/(x*sin x + cos x))² dx
Let x*sin x + cos x = t
=> dt = x*cos x.
Multiply and divide by cos x and in the numerator take (x²)²= (x³)*x
Now rewriting in terms of t and dt:
∫(x³/cos x)*dt/t²
Using integration by parts:
=(x³/cos x)(-1/t) - (x³/cos x)∫(-1/t) dt
=-(x³/cos x)/(x*sin x + cos x) + (x³/cos x)*(ln t) + c
Replacing substituents:
= (x³/cos x)* ln (x*sin x + cos x) - (x³/cos x)/(x*sin x + cos x) + c
❤❤❤❤😊😊 Nice one
do you have any tutorials on differential equations?? i kinda find it confusing sometimes..
just wondering..thanks:)
question :
instead of replacing x with theta ^2 , can't we just change the limits of integration into terms of x , and keep the X as it is ?
BTW , great explanation ^__^
Yes, you can do that, but remember, it's all about making it easier for yourself. For this specific problem, it's much easier to substitute x back in instead of solving for the bounds in terms of theta.
It would be slightly faster to just change the limits of integration. You wouldn't have to rewrite the equation in terms of x. Also, changing the limits of integration is not difficult; plugging in to x= Theta^2 is pretty much what she does at 7:16
your tutorials had helped me a lot!!:)
just a little volume on the audio would make it nonetheless perfect..thanks anyways!!!
You are so awesome. I got the exact tip i needed. You are so pretty. Pretty and intelligent, that's complete
Is there a reason why you didn't change the limits of integration during the x substitution process?
You're right, I should change them to keep them consistent, but most of the time I don't because I'm just going to back-substitute at the end of the problem anyway, and I'll end up changing them right back to what they were originally.
Thanks!
integralCALC if you change the limits of integration you don't need to substitute back because you already changed the limits and then you can just plug in for x
If you don't change the bounds, pls make sure you specify the associated variable. i.e with \theta=.
What??? Aren't cos and thita square a single term??? And thita cube is a orher term so shouldn't we choose between cos thita square and thita cube as to which would be the x substitution?
You could have changed the limits of integration, when you did the x-substitution, to make your life easier, where your integral in terms of x would go from (pi/2) to (pi). Other than that, great.
also tutorials related to engineering surveys..please, let me know if you know some links!!
thanks,i'd appreciate it!!
why is pi always left alone in final answers? since it has a definite value why can't it be simplified?
It was incorrect for her to repeatedly write the original integration limits once the substitution of x = theta squared was made. Each step must be mathematically correct. Either drop the limits and work with an indefinite integral or change the limits to [ pi/2 to pi] in the substituted integral. In fact, it would have been better to just change the limits immediately and then evaluate once the IBP was complete. Would have saved steps and writing all those radicals.
you should try wolframalpha . com. it should give you the step-by-step solution! :D
hahaha you bet!! :)
Thanks aloooot ♡,,that was helpful: )
Please can u help me with this
Integral of cos(x) x e^x dx
Hey! For specific questions like this, try WolframAlpha(dot)com. It will give you step-by-step solutions for this kind of stuff. Hope that helps!! :)
dont u need to change the boundaries when u do the U-sub?
You can, but if you end up back-substituting at the end of the problem to put the integral back in terms of the original variable (like I did), then you can leave the bounds the same, instead of changing them and then changing them back. :)
hey! i don't have any videos on engineering surveys, but i have quite a few on differential equations. check out: integralcalc(dot)com/#differential equations. hope that helps! :)
I need help...with integral...
I thought you had to change the limits since you use u-sub
You have two options to deal with the limits of integration. You can either change them when you make the substitution, and then you'll be able to evaluate using those limits at the end of the problem (this is the method you're suggesting), or you can leave the limits as they are, back-substitute to get the result back in terms of x, and then use the original limits of integration to evaluate at the end of the problem (this is the method I used). Both methods will get you to the same correct answer. :)
I can evaluate it ,,if it was integral of cos(x) e^x dx
You are SO cute ma'am for a teacher I am taking Differential Equations Honors at San Jacinto
Because if you simplify it by writing out the answer you won't get the exact answer unless you write it out to a trillion decimal places.
Standard computation time exceeded...
Sure. But as soon as you change to "x" the LIMITS OF INTEGRATION must reflect this change as well!!!
+John Michael Twist +calculusexpert.com Yeah, don't we need to change the Limits because we made a substitution? The answer is correct though per stewart solution guide. Oh, because we sub back in our original substitution we don't need to touch our limits!
Sean Ashcraft Got to write things consistently. Change the expression, then limits have to be written changed too. This is mathematics.
I agree that she should have changed the limits of integration, in which case, she would have gotten (pi/2) and pi, respectively. However, she did revert back to the original variable, which is another way to do a definite integral by a u-substitution.
Please please help me ♡
your left eye is bigger than the other one