Integrals computation: Help me please

[Phantony]

Hi all,

can you help me to compute these integrals?

$$\int \frac{x \sqrt{a x+b+x^2}}{d^2+x^2} \, dx$$

$$\int \frac{x}{\left(d^2+x^2\right) \sqrt{a x+b+x^2}} \, dx$$

a,b and d are real and positive.

I tried with Mathematica, but the results involve logarithmic functions with complex argument whereas I need a solution in the real domain.

Thank you to all in advance.

 

[Ray Vickson]

Hi all,

can you help me to compute these integrals?

$$\int \frac{x \sqrt{a x+b+x^2}}{d^2+x^2} \, dx$$

$$\int \frac{x}{\left(d^2+x^2\right) \sqrt{a x+b+x^2}} \, dx$$

a,b and d are real and positive.

I tried with Mathematica, but the results involve logarithmic functions with complex argument whereas I need a solution in the real domain.

Thank you to all in advance.

Maple 14 delivers answers in terms of logarithms and square roots and the like. These may be real or complex, depending on the relative magnitudes of a, b and d, but they are written in a symbolic form that does not involve complex quantities explicitly. They are much too large and complicated to be reproduced here. All I can suggest is that you try a tool other than Mathematica; it is often the case that when something does not work out well using one symbolic package, it is better done in another (and there is no consisitency: no one package is universally better than another).

RGV

 

[Phantony]

Thank you soo much, RGV.
I will try with maple and I'll let you know.

:)

 

[Ray Vickson]

Thank you soo much, RGV.
I will try with maple and I'll let you know.

:)

The best way to do it is to recognize that for real d and x we have Re(1/(x + I*d)) = x/(x^2 + d^2), so you can integrate g1 = sqrt(x^2 + ax + b)/(x + I*d) or g2 = 1/(x + I*d)/sqrt(x^2 + ax + b), then (by assuming a,b,x,d real) take the real part *after* doing the integrals, by using the 'evalc' command. This shorter and sweeter than a direct approach.

RGV

 

[Phantony]

Thank you again RGV.

Just one other thing. I'm using maple for the first time and I think that I'm doing something wrong because the result which I obtain is very long.
Is this the right sequence of commands?

$$int( \!{\frac { \sqrt{{x}^{2}+bx+c}}{x+id}},x)$$
$$assume(x, 'real', b, 'real', c, 'real', d, 'real')$$
$$evalc(\%)$$


Phantony.

 

[Ray Vickson]

Thank you again RGV.

Just one other thing. I'm using maple for the first time and I think that I'm doing something wrong because the result which I obtain is very long.
Is this the right sequence of commands?

$$int( \!{\frac { \sqrt{{x}^{2}+bx+c}}{x+id}},x)$$
$$assume(x, 'real', b, 'real', c, 'real', d, 'real')$$
$$evalc(\%)$$


Phantony.

No, those are all wrong. You should use
J:=int(sqrt(x^2+a*x+b)/(x+I*d),x);
J1:=evalc(J) assuming real;
or
J1:=evalc(J) assuming a>0,b>0,d>0,x>0;
If you don't want output echoed on the screen, use the command end ':' instead of ';'

I actually prefer to use something like g1:=sqrt(x^2+a*x+b)/(x+I*d): (or ;) then put
J:=int(g1,x). That way I can get back later to the function g1, or to the function
f1 given as:
evalc(g1) assuming x>0,a>0,b>0,d>0; f1:=Re(%) assuming x>0,a>0,b>0,d>0;
This can all be shortened by setting
assns:=x>0,a>0,b>0,d>0;
evalc(g1) assuming assns; f1:=Re(%) assuming assns.

RGV

 

[Phantony]

I have no words to express my gratitude to you.

I solved the problem

:!):!)