Question

Find \(\int \frac{2 dx}{x^2-64}\).

(a) –\(log⁡\left |\frac{x+8}{x-8}\right |+C\)

(b) \(\frac{3}{2} log⁡\left |\frac{x+8}{x-8}\right |+C\)

(c) \(log⁡\left |\frac{x+8}{x-8}\right |+C\)

(d) \(\frac{1}{8} log⁡\left |\frac{x-8}{x+8}\right |+C\)

The question was asked in my homework.

Question is taken from Integrals of Some Particular Functions topic in chapter Integrals of Mathematics – Class 12

Answer 1

The correct choice is (d) \(\frac{1}{8} log⁡\left |\frac{x-8}{x+8}\right |+C\)

The best explanation: \(\int \frac{2 dx}{x^2-64}=2\int \frac{dx}{x^2-8^2}\)

By using the formula \(\int \frac{dx}{x^2-a^2}=\frac{1}{2a} log⁡|\frac{x-a}{x+a}|+C\)

∴\(2\int \frac{dx}{x^2-8^2}=2(\frac{1}{(2(8))} log⁡|\frac{x-8}{x+8}|)+2C_1\)

=\(\frac{1}{8} log⁡|\frac{x-8}{x+8}|+C\)