Proof-1-a-2-x-2-dx-1-2a-ln-a-x-a-x-c-

Question Number 27271 by GANGADHARSETHI last updated on 04/Jan/18

P r o o f

\int \frac{1}{a^{2} - x^{2}} d x = \frac{1}{2 a} l n ∣ \frac{a + x}{a - x} ∣ + c

Answered by sma3l2996 last updated on 04/Jan/18

\int \frac{d x}{a^{2} - x^{2}} = \int \frac{d x}{(a - x) (a + x)}

\frac{1}{(x + a) (a - x)} = \frac{α}{a + x} + \frac{β}{a - x}; α = \frac{1}{2 a} = β

s o : \int \frac{d x}{a^{2} - x^{2}} = \frac{1}{2 a} \int (\frac{1}{a + x} + \frac{1}{a - x}) d x

= \frac{1}{2 a} (l n ∣ a + x ∣ - l n ∣ a - x ∣) + c

= \frac{1}{2 a} l n ∣ \frac{a + x}{a - x} ∣ + c