Integrating with a root in the numerator:

[burt]

I was given the following problem to integrate [MATH]\int\frac{\sqrt{x+4}}{x(x+5)}\ dx[/MATH].
My problem is that I don't know how to deal with that roo in the numerator. My instinct is to do a u-substitution and then partial fractions, but I do not see a substitution that can be made. Is there another method I should be thinking of? Or, is there something to substitute that I am missing?

 

[Deleted member 4993]

I was given the following problem to integrate [MATH]\int\frac{\sqrt{x+4}}{x(x+5)}/ dx[/MATH].
My problem is that I don't know how to deal with that roo in the numerator. My instinct is to do a u-substitution and then partial fractions, but I do not see a substitution that can be made. Is there another method I should be thinking of? Or, is there something to substitute that I am missing?

I have not tried it yet, but:

x = 4 * tan²(t) ......... might work

WA says it will work but the answer will involve tanh function - that too at half-angle!!

I am not going after that monster!!

 

[pka]

I was given the following problem to integrate [MATH]\int\frac{\sqrt{x+4}}{x(x+5)}\ dx[/MATH].
My problem is that I don't know how to deal with that roo in the numerator. My instinct is to do a u-substitution and then partial fractions, but I do not see a substitution that can be made. Is there another method I should be thinking of? Or, is there something to substitute that I am missing?

This is a beast of an integral. Here is a complete solution. You may want give it a try before look at the solution.

 

[burt]

@pka @Subhotosh Khan
Thanks! It actually seems there is a simpler solution which can be found by making [MATH]u=\sqrt{x+4}[/MATH] which makes [MATH]x=u^2-4[/MATH]. This method gets

[MATH]\frac{2}{5}\ln\bigg|\frac{\sqrt{x+4}-2}{\sqrt{x+4}+2}\bigg| +\frac{2}{5} arctan\,(\sqrt{x+4})+C[/MATH]