❖ Trigonometric Substitution : Definite Integral ❖

Trigonometric Substitution : Definite Integral

In this video, I explore a basic example of trigonometric substitution in calculus. Trigonometric substitution is a powerful technique used to simplify integrals, but it requires different substitutions based on the expression involved.
Note that in this case since we have a definite integral, we do not need to worry about making a triangle as in the case of an indefinite integral requiring trigonometric substitution.

🔍 What You’ll Learn:

The concept of trigonometric substitution and when to use it.

Different types of substitutions for various expressions.

How to label the right triangle from our initial substitution and its importance.

A complete example demonstrating the substitution process and integration.

Tips for handling tricky integrals that may require additional techniques after substitution.

This tutorial is perfect for students looking to deepen their understanding of integration techniques in calculus. By the end of this video, you'll be equipped to tackle trigonometric substitution problems with confidence.

Don’t forget to like, subscribe, and hit the notification bell for more math tutorials and calculus strategies!

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[graceface]

thank you again! you are the sole reason I am surviving Calc 2.

[grounded]

Thank you so much! No one else would explain how to convert the limits!

[ihearthikary]

Those conversion for limit of integrals was all I need and you showed them very clear, thanks so much! :D

[cinecrime]

Thanks dude, your videos are clear and easy to understand!

[bassionbean]

You just need to know that when a function looks like x2 + a, the best substitution is btan(theta). In this case b = 2 because you need to factorise it, and so the square of it needs to equal the a value, which here is 4. If it were x2 + 9, then the substitution would be 3tan(theta)

[fionaNESS]

@mathforphysics
if you look at his video on the top right, the trig subs video, you'll see that for these different types of integrals (they look really familiar) you have to do a trig substitution, and depending on the form of that integral you either sub in asinx, asecx or atanx.
so no, noone would guess to set x as 2tanx :)

just look at his other video

[saadb]

hey PatrickJMT. i used the following sub: let definite integral came to be: [arcsinh(x/2)]....plug and chug 2 and 0 as the limits....use the exponential form of sinh...got exactly the same answer= ln(1+sqrt2)

[imdahero]

thanks for the videos keep up the good work ;)

[MrJLivermore]

how do you know when to change the limits of integration and when do you leave them the same?

[tiagobarbosa]

patrickJMT in the minute 3:30 secant of zero is 2/(4)^1/2, don't you think?

[thisislorelai]

What if there’s no value of arctan in unit circle?? I can’t find any tutorial about it😢

[ExpHPKun]

What bothers me is figuring out the correct limits of integration.  In this problem, tan(-3π/4)=1, but if you used that as your upper limit, it would change your final result.

I suppose one could argue that arctan is defined with a range of (-π/2, π/2), but... isn't that just convention?

The issue is even more profound with functions like sqrt(1-x^2).  After doing sine-substitution, your final result will have a term of (1/2)t.  This means you can't even have one of the limits change by 2π!

[vimrivera]

***sec pi/4 is sq.root of 2. Answer should be ln |sqroot2 + 1|.

[nikeshnikku]

man
since tan 45=1
that implies value of tan pi/4 in radians is 1

[jinwei]

Can I just use the trigo substitution and get the ln|answer| back in terms of x, then only sub in the 2 and 0 into the equation to get the final answer instead of changing the 2 and 0 to Pie/4?

[beme]

Yes you're the same, I mean I love his voice, I can't be possibly wrong I guess

[PureInsanity]

Thank you!
and happy new Jewish year.

[TheFsadfsd]

The dislike bar is the size of the joint I'm smoking!

[MMIST]

Why sectheta not equal to 1/costheta but ln |sectheta+tantheta)???? How do i know what law to use?

[danecarlson]

can someone explain to me how on earth the first 15 seconds happen? where does x = 2tan theta???