I know I can solve this with u sub but I dont really like solving stuff like this with it since it does not always work , so are these 2 methods correct is the most important thing that I want to know
#solving the integral best method
past axle
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I know I can solve this with u sub but I dont really like solving stuff like thi...
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past axle
like I dont think I can solve this with u sub integral(dx/x+5)
small skiff
Not sure what you're doing there.
u = 1 + x^5, du = 5x^4 dx
x^4 dx/(1 + x^5) = (1/5)5x^4 dx/(1 + x^5) = (1/5)du/u
And this is easy to integrate.
past axle
how do you choose u ? why did you take 1+x^5 in this case ?
past axle
like I dont think I can solve this with u sub integral(dx/x+5)
and how would you solve this ? you cant use u sub , so you would have to use my method
at least that's what is looks like
oh you could actually
I guess the method I was shown is just terrible
yeah its trash I could not even solve it if I have stuff like ln(x) or sin(x) or stuff like that
I mean its not trash it might be easier ? idk
lean sundial
When you see a function and its derivative, that usually means you can use a U substitution
Since U-sub is essentially reverse chain rule
You choose the entire denominator (1+x⁵) because the derivative will leave you with 5x⁴, which you have an x⁴ in the numerator, then factor the 1/5 out
past axle
Since U-sub is essentially reverse chain rule
yeah I got what u god ,bdw if I have arct ln(x) I would use u sub ?or integration by parts
I know I have to use them when I have stuff like that in the integral but which is better ?
and what should I put n as if I hvae them since I know what to do if I have normal stuff like () e^() sqrt() 1/() 1/sqrt() 1/()^2 n should most of the time be whats in ()
past axle
yeah I got what u god ,bdw if I have arct ln(x) I would use u sub ?or integrati...
but what do I do if I have stuff like this ?
lean sundial
I haven't come across a problem like that so I'd have to think about it
small skiff
yeah I got what u god ,bdw if I have arct ln(x) I would use u sub ?or integrati...
Integration is something that just gets better with practice and experience. Not all substitutions are easy to explain, so some need to be just memorized.
In your case, let's try integration by parts first.
u = arctan(ln(x)), dv = dx
du = dx/(x(1 + ln(x)^2)), v = x
∫(arctan(ln(x))dx) = x arctan(ln(x)) - ∫(dx/(1 + ln(x)^2))
The last one probably isn't expressible in elementary functions. Maybe it is with Ei(x) or something, though.