How to compute antiderivative in pic?

[glacial axle]

Title

I was thinking of doing the u' times u formula

but then arctg would have -1 as exponent which makes that formula not applicable

So what do?

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[timid cloak]

I recommend using u-sub

try u=2tan(x)

things simplify very well

[trim inlet]

What is "arctg"? Do you mean "arctan"?

[glacial axle]

yea

[timid cloak]

arctg is also valid

same thing with sin and sen

[trim inlet]

It's not a notation I've ever seen.

[glacial axle]

EUbros represent

[timid cloak]

yes

@glacial axle did u try that u-sub?

[glacial axle]

I dont think we're supposed to do u-sub for these exercises, I think these are supposed to have a simpler solution as they're more introductory also I haven't done much u-sub before so I'm not super confident with it

[timid cloak]

oh sure

[trim inlet]

Then isn't it also possible that this is supposed to be a problem where you practice u sub?

[timid cloak]

oh yh

I just saw what u have to do

@glacial axle try computing the derivative of arctg(x/2)

see what u get

you'll see a familiar expression

[trim inlet]

Holy shit, I'm an idiot. I already did that, but I did it wrong.

[glacial axle]

But if I separate it into two fractions (one multiplied by the other) and make the first fraction be the derivative of the second I still have the problem that the exponent of the second fraction would be -1 which makes the rule invalid.

You're trying to get to this form, right?

[timid cloak]

wdym?

nop

there's another form

lemme write it down

$\int { \frac{f^{'}(x)}{f(x)} dx} = ln(f(x)) + C$

[livid frigate]

Arm

[timid cloak]

it's really the absolute value, but u get the idea

[glacial axle]

I'm sorry, but I don't see how to get to that form... 😦

[timid cloak]

what's the derivative of ln(f(x))?

[glacial axle]

f'(x)/f(x)

[timid cloak]

yes

now integrate both sides

do the anti derivative of both sides

[glacial axle]

f(x)/int(f(x))

[timid cloak]

?

[glacial axle]

Not sure what you mean both sides

[timid cloak]

u just get the integral of f'(x)/f(x) = ln(f(x) + c

[glacial axle]

ok and...?

[timid cloak]

so that's the form for n=-1

but still

u can do this without using it

but yh, this is the easiest way

and probably the intended way hahah

[hardy bronze]

Let
U = arctan(x/2)*1/2
Du = 1/4 + x^2

$2\int \frac{du}{u} = 2\ln{|u|} + C= 2\ln{|\frac{1}{2}\arctan(\frac{x}{2})|} + C$