Question

Find $\int \frac{dx}{x^2-a^2}$ and hence evaluate $\int \frac{dx}{x^2-25}$.

Answer 1

$\int \frac{dx}{x^2-a^2}$

$=\int \frac{dx}{(x-a)(x+a)}$

$=\frac{1}{2a}\int \frac{((x+a)-(x-a))dx}{(x-a)(x+a)}$

$=\frac{1}{2a}\int \frac{(x+a)dx}{(x-a)(x+a)}-\int \frac{(x-a)dx}{(x-a)(x+a)}$

$=\frac{1}{2a}\int \frac{dx}{(x-a)}-\int \frac{dx}{(x+a)}$

$=\frac{1}{2a}[\log |x-a|-\log |x+a|]+c$

$=\frac{1}{2a}\log \left|\frac{x-a}{x+a}\right|+c$

Now $\int \frac{dx}{x^2-25}$

$=\int \frac{dx}{x^2-5^2}$

$=\frac{1}{2\times 5}\log \left|\frac{x-5}{x+5}\right|+c$

$=\frac{1}{10}\log \left|\frac{x-5}{x+5}\right|+c$