Integral containing something over the root of x²-a²

[Telemachus]

Homework Statement

Evaluate the next integrals, expressing it previously in the forms that contains \sqrt[ ]{a^2+u^2}, \sqrt[ ]{a^2-u^2} and \sqrt[ ]{u^2-a^2}, and then solve it using the integral table;

The exercise which I couldn't solve:

\displaystyle\int_{}^{}\displaystyle\frac{x+3}{\sqrt[ ]{x^2+2x}}dx

The Attempt at a Solution

I started by completing the square:

x^2+2x=(x+1)^2-1

Then

u^2=(x+1)^2
u=(x+1)\Rightarrow{x+3=u+2}
du=dx

\displaystyle\int_{}^{}\displaystyle\frac{x+3}{\sqrt[ ]{x^2+2x}}dx=\displaystyle\int_{}^{}\displaystyle\frac{u+2}{\sqrt[ ]{u^2-1}}du

From here I've tried to solve it by parts
t=u+2
dt=du

dv=\displaystyle\frac{1}{\sqrt[ ]{u^2-1}}du
v=\ln|u+\sqrt[ ]{u^2-1}|

\displaystyle\int_{}^{}\displaystyle\frac{u+2}{\sqrt[ ]{u^2-1}}du=(u+2)\ln|u+\sqrt[ ]{u^2-1}|-\displaystyle\int_{}^{}\ln|u+\sqrt[ ]{u^2-1}|du

If I try again by parts with -\displaystyle\int_{}^{}\ln|u+\sqrt[ ]{u^2-1}|du it gets more complicated.

So, what do you say?

Bye there.

 

[Dick]

You can do integral u/sqrt(u^2-1) with a simple substitution. So split (u+2)/sqrt(u^2-1) up into two integrals.

 

[hunt_mat]

The integral
<br /> \int\frac{u+2}{\sqrt{u^{2}-1}}du<br />
Can be computed by the substitution
<br /> u=\cosh v<br />
Everything should come out in the wash then

 

[Telemachus]

Thank you both.

This is what I wasn't seeing:

You can do integral u/sqrt(u^2-1) with a simple substitution. So split (u+2)/sqrt(u^2-1) up into two integrals.

Bye there!

 

[Telemachus]

I've arrived to this solution:

\displaystyle\int_{}^{}\displaystyle\frac{x+3}{\sqrt[ ]{x^2+2x}}dx=\sqrt[ ]{(x+1)^2-1}+2\ln|x+1+\sqrt[ ]{(x+1)^2-1}|+C

 

[Dick]

I've arrived to this solution:

\displaystyle\int_{}^{}\displaystyle\frac{x+3}{\sqrt[ ]{x^2+2x}}dx=\sqrt[ ]{(x+1)^2-1}+2\ln|x+1+\sqrt[ ]{(x+1)^2-1}|+C

Seems ok to me. I would write x^2+2*x instead of (x+1)^2-1. But that's up to you.

 

[Telemachus]

Thanks Dick. BTW nice nickname :D

 

[gomunkul51]

yup, got the same answer !

P.S: anyone knows a program that writes LaTeX and has an intuitive interface? :)

 

[nekronaut]

yup, got the same answer !

P.S: anyone knows a program that writes LaTeX and has an intuitive interface? :)

Try this site, I use it frequently :)

http://www.codecogs.com/latex/eqneditor.php

 

[gomunkul51]

Try this site, I use it frequently :)

http://www.codecogs.com/latex/eqneditor.php

Thank you ! :)