Integration using t=tanx

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In this video, I showed how to integrate a rational function using the t=tanx substitution

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... Good day Newton, I have just one tiny comment at time 8:48: Why the u^2 + u in the denominator of the integrand instead of u^2 + 1? ... I know it is accidentally written ... Great presentation from your hand as always; I like these integrals full of surprises ... Take care, Jan-W

jan-willemreens

Learning from you. I would suggest others to try solving the question by simplifying firstly the expression using the identity (sin^2+cos^2=1 --> sin^2 = 1-cos^2). I think this will save time.

AbdelRahmanAllan

i had no idea to attempt this integral, thanks mr prime newtons, more blessings

nicholasbarinda

ur explaination is the best ever i thank u for everything u do
thank u for ur help ❤❤

AliMohamedsayed

it was good the way you made things look easy but always start by showing us the reason why we choose to use that method of intergration

B.r.y.t.o.n

Hi, it's a magical task, the derivation of the first part is great. If we multiply by 1/(cosx)^2, we get (1/2) int (secx^2/(tanx^2+2))dx, and let t=tanx, (1/2)int (1/( t^2+2))dt and we can already write + C, Thank's

prollysine

At the first step we should to simplify 2(sinx)^+4(cosx)^2. It is equal to 2+2(cosx)^2. After that we can use t=tgx, as it is shown.

DmitryKrechet

It is tricky but you are explaining it very very good

XolaMbusi

Why cant we write the int{1/(2sin^x+4cos^2x)dx} as ln of the inside thing directly

srisaishravan

I don't think that approach is necessary... Maybe.

When i did Calculus it was well known that tan'(x)=1/cos^2(x). We observe that all the trigonometric finctions in the denominator share the same power and forcefully pull out cosx as common factor obtaining :
1/cos^2(x)*1/(2tan(x)+4)dx
We observe we have the derivative of a function composition and the answer is instantaneous when considering arctan'(x/a)=a/(x^2+a^2)

Wonder how i could explain this better... I think this will be a bit confusing for the less experienced. When i did my courses i completed all the problems in the textbook and then a few more as i found calculus entertaining

dan-florinchereches

Thank you sir, but I have a question at 8:08
why is it sqrt2 and not 2

slowwindow

In minute 8:47 your wrote u instead of 1 in the denominator

varhaforsson

that triangle is realy good way to get sin and cos my doctor didnt use that

crtlallaw

what if i have in denominator tan or cot ?

vismof

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