Question

What is the value of the integral \(\int_0^∞ \frac{1}{c^x}  dx\)?

(a) \(\frac{1}{logc} \)

(b) \(\frac{2}{logc} \)

(c) \(\frac{\pi}{logc} \)

(d) \(\frac{1}{2logc} \)

I got this question in examination.

This is a very interesting question from Special Functions topic in chapter Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Right choice is (a) \(\frac{1}{logc} \)

For explanation I would say: \(\int_0^∞ \frac{1}{c^x}  dx \)

= \(\int_0^∞ e^{-xlogc}  dx \)

Substitute \(xlogc = t \)

\(logc \,dx = dt \)

\(= \int_0^∞ e^{-t}   \frac{dt}{logc} \)

\(= \frac{1}{logc} * \int_0^∞ e^{-t}  dt \)

\(=\frac{1}{logc}. \)