Trigonometric substitution with tangent (KristaKingMath)

Learn how to use trigonometric substitution with a tangent substitution to evaluate an integral.

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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?!”

[kristakingmath]

Because sqrt(9+16x^2) is not p/m 3+4x. If you square 3+4x, you get 9+24x+16x^2. Hope that helps clarify it!! :)

[holyshit]

sqrt(ax^2+bx+c)=u-sqrt(a)x a>0
sqrt(ax^2+bx+c)=(x-x_1)u b^2-4ac>0

These sustitutions should be enough but there is one more substistution

sqrt(ax^2+bx+c)=xu+sqrt(c) c>0

These substitutions can be applied to the integral of trig functions

Rewrite trig function with secant and tagent then substitute
sec(x)=u-tan(x)
Rewrite trig function with cosine and sine then substitute
cos(x)=(1-sin(x))u

[moinaction]

I so much like your teaching style. You don't just steps like many lecturers do and that makes it easier to understand you a lot. Thanks for doing what you do best

[ungell]

That is a mind opening problem! Thank You :)

[CusPedro]

After watching this video, I'm going to drop the calculus 2 course I'm currently taking.

[MrKidda]

hi first of all i want to say that your videos are really helpful so thanks. secondly, i'm little confuse by the annotation. the annotation says that the original integral suppose to be 9/4 int sec^2 theta but didn't you added 9/4 int sec^2 theta in order to get 18/4 int of sec^2 theta. i could be wrong, but could you explain it quickly that how did you get original integral to be 9/4. like annotation says. Thanks!

[zakmatew]

I just hope my teacher doesn't give us this type of problem on the test. It took you so much time to solve it. Yes, you had to explain the process which added some time but still this problem takes like 10-15 minutes to solve.

[Sandra-nvwm]

This was so helpful! Thank you so so so much! It is often so hard finding a good tutorial but this seems like you did it today and not 4 years ago!

[kristakingmath]

@williampayson there's already an annotation to represent that, but thanks for the feedback

[hg.]

Krista, at min 13:43 you take the Integral of a Secant which is a Natural Log. Can you point me to one of your videos that explains this type if integration?

[hg.]

Thank you Krista for taking time out to remind us about how Integration by Parts works (!). By the same token I could have used a mini-refresher on where the "ln" came from in the last lap of the problem, but I vaguely remember you have videos on that too. Onward and :)

[sabriath]

Well....I mean, that's the long way to do it, otherwise you could just memorize the following:

∫√(a+bx^2) dx = x√(a+bx^2)/2 +

Then just plug in your values....done.

[diannak]

I don't understand your note at 14:03... The (9/4 on right) * (1/2) becomes (9/8), and it simplifies to something different..? 

[MsReeses]

im confused on how you got ln|sec theta + tan theta|. Can you please explain? thanks

[BaconJunkee]

Awesome! I feel so ready for my quiz today :D I wish you were my teacher!

[cosmickitty]

why is the dx term (sec) squared though? Wouldn't it just be 3/4sec theta, and not squared?

[foodparadise]

To easy the complication I would use the sec^3 d du formula from 9/4 integral sign sec^3 theta

[Chae_shay]

Best I've found on Youtube so far. I think I'll rest a bit better tonight in preparation for my quiz tomorrow. Thanks so much! Don't stop making videos! I see you've made a lot already which makes me happy! Subscribed!

P.S. You have a very nurturing voice :]

[MrYaseen]

great vid again! but where did the 9/4sec(theta)tan(theta) come fro on the left at 12:48? i never knew we had make integrals equal each other and stuff...is there a method that works without doing that? also, wouldnt multiplying everything by 1/2 as you put in the annotation make 1/2*9/4=9/8? i dont see 1/2 anywhere...

[thevegg]

Thanks for your videos. Quick question. I saw a video where someone solved the integral of r^3*dx/sqrt (4+r^2) using u-substitution but I thought you said this must be done by trig substitution ice it has a radical and perfect squares underneath. If it can be done using u-substitution then why do we ever need trig substitution for this type of equation?