Question

When solved by the method of Differentiation for the given integral i.e \(\int_0^∞ \frac{x^{2}-1}{log⁡x} dx\) the result obtained is given by _______

(a) log⁡4

(b) log⁡3

(c) 2log⁡3

(d) log⁡8

The question was posed to me in class test.

This key question is from Differentiation Under Integral Sign in section Partial Differentiation of Engineering Mathematics

Answer 1

The correct option is (b) log⁡3

To explain: To solve this problem let us assume the given function is dependent on α

Such that α=2 & thus \(f (α) = \int_0^∞ \frac{x^{α}-1}{log⁡x} dx\)

\(f’(α) = \int_0^1\frac{∂}{∂α} \left(\frac{x^{α}-1}{log⁡x}\right)dx\) …..Leibnitz rule

\( =\int_0^1\frac{x^α.log⁡x}{log⁡x} dx\)

\( = \int_0^1 x^α  dx = [\frac{x^{α+1}}{α+1}]_0^1 = \frac{1}{α+1}\)

We have \(f’(α) = \frac{1}{α+1}\)

Thus \(f (α) = \int\frac{1}{α+1}dα+c\)

f (α) = log(α+1)+c

or f (α) = log(α+1) …… neglecting constant since the function is assumed

thus f (2) = log(2+1) = log(3).