Question

When solved by the method of Differentiation for the given integral i.e. \(\int_0^1 \frac{x^2-1}{log_2⁡x}  dx \) the result obtained is given by _________

(a) log⁡5

(b) 3 log 3

(c) log 4

(d) 2 log 3

I have been asked this question in an interview.

I want to ask this question from Differentiation Under Integral Sign in division Partial Differentiation of Engineering Mathematics

Answer 1

Right answer is (a) log⁡5

The best I can explain: \(\int_0^1 \frac{x^2-1}{log_2⁡x}  dx\) can also be written as \((log⁡2 \int_0^1 \frac{x^2-1}{log ⁡x} dx)…….(1).\)

Here during integration changing \(log_2⁡x = \frac{log⁡x}{log⁡2}\)  and substituting we get (1) since  logarithm to base ‘e’ can be easily integratable.

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

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

\(f’(α) = log⁡2 \int_0^1\frac{∂}{∂α} \left(\frac{x^{α}-1}{log⁡x}\right)dx\) …..Leibniz rule

 \( = log⁡2 \int_0^1\frac{x^α.log⁡x}{log⁡x}  dx\)

\( = log⁡2 \int_0^1 x^α  dx = [\frac{x^{α+1}}{α+1}]_0^1 = \frac{1}{α+1}\)

We have \(f’(α) = log⁡2.\frac{1}{α+1}\)

Thus \(f (α) = log⁡2\int\frac{1}{α+1}dα+c\)

f (α) = log⁡2 .log(α+1)+c

or f (α) = log(α+1+2) neglecting constant since the function is  assumed

thus f (2) = log(2+3) = log(5).