Question

Find \(\int \frac{8 dx}{x^2-16}\).

(a) \(log⁡\left |\frac{4+x}{4-x}\right |+C\)

(b) –\(log⁡\left |\frac{4+x}{4-x}\right |+C\)

(c) \(8 log⁡\left |\frac{4+x}{4-x}\right |+C\)

(d) \(\frac{1}{8} log⁡\left |\frac{4+x}{4-x}\right |+C\)

The question was asked in class test.

Origin of the question is Integrals of Some Particular Functions in chapter Integrals of Mathematics – Class 12

Answer 1

Correct option is (a) \(log⁡\left |\frac{4+x}{4-x}\right |+C\)

To elaborate: \(\int \frac{8dx}{16-x^2}=8\int \frac{dx}{4^2-x^2}\)

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

∴\(8\int \frac{dx}{4^2-x^2}=8(\frac{1}{2(4)} log⁡\left |\frac{4+x}{4-x}\right |)+8C_1\)

\(8\int \frac{dx}{4^2-x^2}=log⁡\left |\frac{4+x}{4-x}\right |+C\)