LCKurtz said:The derivative of ##\arctan(x^{-\frac 1 2})## is$$
\frac 1 {1+\frac 1 x}\left(-\frac 1 2 x^{-\frac 3 2}\right )=
-\frac 1 2 \frac{x^{-\frac 1 2}}{x+1}$$That might help your integration by parts.
ruiwp13 said:Homework Statement
Solve the following integral:
∫arctg(1/√x)dx
Homework Equations
∫u'/1+u^2 = arctg(u)+c
∫u.v' = u.v - ∫ u'.v
The Attempt at a Solution
So , I tried to define u as arctg(1/√x) but I'm having trouble finding the du. The derivate of arctg(1/√x) is -1/2*x^(3/2), so it stays -1/2*x^(3/2)/1-(1/√x)^2 (converting to the derivate u'/1+u^2) ? I was thinking about getting the derivate of arctg(1/√x) and doing this by parts. dv = dx , u = arctg(1/√x)
Substitute for √x and use partial fractions.ruiwp13 said:yes, I got this x.arctg(1/√x)+∫x/sqrt(x)*(2x+2) but now I'm having trouble on solving the integral :p any help?
It's that thing where you have one polynomial divided by another, P(x)/Q(x). You rewrite it as a sum of simpler fractions. http://en.wikipedia.org/wiki/Partial_fractions_in_integrationruiwp13 said:I don't remember how to use the partial fractions , it's that thing with A/(x-1) * B/(x+1) and A*(x+1)*B(x-1) thing?
As I indicated, you first need to get rid of the surd by substituting a different variable for √x.ruiwp13 said:I transformed x/sqrt(x)*(2x+2) into sqrt(x)/2x+2 .
An undefined integral is an integral that does not have a specific value. This can occur when the function being integrated is undefined at certain points, or when the limits of integration are not well-defined.
Arctan, or inverse tangent, is used to solve undefined integrals because it allows us to convert a complex integral into a simpler one. By using the trigonometric identity arctan(x) = arccos(1/x), we can rewrite integrals involving square roots or inverse trigonometric functions as integrals involving arctan.
Integration by parts is a method used to evaluate integrals by breaking them down into simpler integrals. It involves using the product rule of differentiation to rewrite the original integral in a different form, which may be easier to solve.
Integration by parts can help solve undefined integrals by breaking them down into simpler integrals that can be evaluated using known techniques. By using integration by parts, we can often transform an undefined integral into one that is well-defined and can be solved using standard integration methods.
While arctan and integration by parts can be powerful tools for solving undefined integrals, they may not always be applicable. Some integrals may be too complex to be solved using these methods, or may require alternative techniques such as substitution or partial fractions.