How to evaluate the above integration. Kindly help me out.

Question

How to integrate $$\int \frac{77\ dx}{(x^2+(x^2)^{1/2}-66)} \tag 1$$ I am trying this question by taking $x^2$ out of the square root sign. Now $x^2$ will become $|x|$. So, the final expression will become $$\int \frac{77\ dx}{x^2+|x|-66}\tag 2$$ Now I am facing problem to integrate this expression further. I can't proceed further. The only thing I have thought is that $|x|$ can be written as $x\cdot(\text{sgn}(x))$. Please help me out.

Answer 1

There are two possibilities:

• $x \le 0 => x^2+|x|-66 = x^2-x-66$
• $x \ge 0 => x^2+|x|-66 = x^2+x-66$

In both cases, you have a second-degree polynomial with a discriminant larger than zero ($265$ to be exact), so you can write them as $(x - a) \cdot (x - b)$.

Then you just use "partial fractions" for solving the integral.