Question

Find \(\int \frac{7dx}{x^2-9}\).

(a) \(\frac{7}{6} log⁡|\frac{x-9}{x+9}|+C\)

(b) \(\frac{7}{9} log⁡|\frac{x-3}{x+3}|+C\)

(c) –\(\frac{7}{6} log⁡|\frac{x+3}{x-3}|+C\)

(d) \(\frac{7}{6} log⁡|\frac{x-3}{x+3}|+C\)

I have been asked this question during an interview for a job.

My query is from Integrals of Some Particular Functions topic in section Integrals of Mathematics – Class 12

Answer 1

The correct answer is (d) \(\frac{7}{6} log⁡|\frac{x-3}{x+3}|+C\)

Explanation: \(\int  \frac{7dx}{x^2-9}=2\int \frac{7dx}{x^2-3^2}\)

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

∴\(7\int \frac{dx}{x^2-3^2}=7(\frac{1}{2(3)} log⁡|\frac{x-3}{x+3}|)+7C_1\)

=\(\frac{7}{6} log⁡|\frac{x-3}{x+3}|+C\)