Trigonometric substitution with sine (KristaKingMath)

Learn how to use trigonometric substitution with a sine 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?!”

[MK-kqmi]

What I like about your videos is that you don't do the most basic problems . You show longer more complicated problems. That's good because most teachers are not going to give you the basic problems in a college course . Thank you again You are doing excellent work. I appreciate your videos!!!

[joshcicchini]

Instead of expanding the integrand to 1 - cos^2(θ), another method would be to employ the fact that sin^2(θ) also equals 1/2 (1 - cos(2θ)). The antiderivative of 25/2 (1 - cos(2θ)) dθ is 25/2 (θ - 1/2 sin(2θ)) + C, which, after applying the double-angle formula sin(2θ) = 2 sin(θ) cos(θ), becomes 25/2 (θ - sin(θ) cos(θ)) + C.

[joshcicchini]

From there, we can quickly insert the earlier-established information from our reference triangle: sin(θ) quite simply is opposite over hypotenuse, or x/5; cos(θ) is adjacent over hypotenuse or √(25 - x^2) / 5. No need to sub the value of θ = arcsin(x/5) back into these, which necessitates extra chicken-scratch and, in the latter case, forms a clunky and perhaps unintuitive nested trig function.

[adamwajih]

it's been three years since I watched your videos and you've been a great source of help for me 😁

thank you Krista for all the work you do !!!

[miguelsajche]

my brain just exploded into millions of pieces

[mathteacher]

Great job! - draw the triangle, it helps

[ycng]

You make calculus easy to learn, Thank You.

[grindsession]

@ 13:40 you plugged in "arcsin (x/5)" for "theta" in "sin(2 theta)" and then used the double angle formula for sin.
If you want to skip 5 minutes of work, 13:40 through 18:10, use the double angle formula first, getting "2 sin(theta) cos(theta)", your triangle shows you what "sin(theta" and "cos(theta)" are so you can plug that in simplify and your done.

[salehahmat]

really I like it so much. You explain very clearly. besides that you give also their applications in our reel life

[kristakingmath]

yes you do! it just takes a little practice. :) it'll be simple one day for you, i promise!! :D

[salehahmat]

Thank you so much for your great explanation

[holyshit]

Suppose we have right triangle
We label one of acute angles theta, opposite side we label as x and label hypotenuse as a
We bisect other of acute angle and calculate cotangent of bisected angle
What we will get ? (a proper u substitution)

[shadowzabyss]

on 16.45 of this video, you mention about the table for simplify those terms. I would like to see it.

[mfundothwala]

Good video simple and straight forward l like it, love you

[Cutieflor]

you are great in every way. i get this now.

[kristakingmath]

i've never heard of a professor require that you know how to generate the identities, so i would say your safe just to remember trig sub as a method for solving integrals. but if you're in a class right now, i'd ask the teacher just to be safe! :)

[kristakingmath]

Good catch, thank you! I added an annotation. :)

[saif]

woahhh, ok so I've only taken precal and decided to just see what this is like aaaand..
O.O
I was just wondering:
do you eventually reach a point where this feels simple?

[sapiegelo]

Very clear explanation. Can you tell me what computer application did you use to explain this?

[irfaniskandar]

That was really long problem but thank you so much! I learned a lot.