Evaluate $\int \frac{dx}{\left(\frac{1}{x}-\frac{1}{a}\right)}$

Question

Evaluate the following integral: $$\displaystyle \int\dfrac{dx}{\left(\dfrac{1}{x}-\dfrac{1}{a}\right)}$$ Where $a$ is an arbitrary constant.

How do I solve this?

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I tried the substitution $$x=a\cos \theta$$

But that got me here (i.e. no where): $$\displaystyle a^{3/2}\int\dfrac{-\sqrt {\cos\theta}.\sin\theta.d\theta}{\sqrt{1-\cos\theta}}$$

How do I simplify this further?

Answer 1

$$\int\dfrac{dx}{\left(\dfrac{1}{x}-\dfrac{1}{a}\right)}=\int\dfrac{ax\,dx}{a-x}.$$ Note that $$\dfrac{ax}{a-x}=\dfrac{(-a)(a-x-a)}{a-x}.$$

Answer 2

Use: $\frac 1 {\frac 1 x -\frac 1 a} =\frac {a^{2}} { a-x} -a$.

Answer 3

Note that the integrand is a rational function which can be easily split: $$\dfrac{1}{\left(\dfrac{1}{x}-\dfrac{1}{a}\right)}= -\frac{ax}{x-a}=-\frac{a(x-a+a)}{x-a}=-a-\frac{a^2}{x-a}.$$ Can you take it from here?

Answer 4

$$I=a\int\frac{xdx}{a-x}=a\int\frac{a-(a-x)dx}{a-x}=-a^{2}\ln\left| a-x \right|-ax+c$$