Solving Integral: x + 4a + b / [x - (a + b)]^2 + c^2

[sccv]

I have to solve this integral

$$\int{\frac{x + 4a + b}{[x - (a + b)]^2 + c^2}}dx$$

where a, b, c are constant

Could anybody know how to solve it ?

 

[learningphysics]

You can rewrite the integral as:

$$\int{\frac{x - (a + b)}{[x - (a + b)]^2 + c^2}}dx + \int{\frac{5a+2b}{[x - (a + b)]^2 + c^2}}dx$$

Can you solve it now looking at the two integrals separately? Do you have integral tables to work with?

 

[dextercioby]

You can rewrite the integral as:

$$\int{\frac{x - (a + b)}{[x - (a + b)]^2 + c^2}}dx + \int{\frac{5a+2b}{[x - (a + b)]^2 + c^2}}dx$$

Can you solve it now looking at the two integrals separately? Do you have integral tables to work with?

He doesn't need tables so solve this kind of integrals.Just well made substitutions.
Your integral should be put in the form:
$$\frac{1}{2}\int\frac{d[[x - (a + b)]^2+c^2]}{[x - (a + b)]^2 + c^2}+ (5a+2b)\int \frac{d[x-(a+b)]}{[x - (a + b)]^2 + c^2}$$

Do u see some patterns for substitutions which should bring the 2 integrals to familiar form??ln & artan in the final result??

Daniel.

 

[sccv]

Thank you!
I also came to get those result.