Question

Given \(f (a) = \int_a^{a^2} \frac{sin⁡ax}{x} dx \) what is the value of f’(a)?

(a) \(\frac{sin⁡3a}{a}\)

(b) \(\frac{3 sin⁡ a^3 – 2 sin a^2}{a}\)

(c) \(\frac{3 sin a^2 – 4 sin a}{a}\)

(d) \(\frac{3 sin a^3 – 3 sin^2 a}{6a}\)

This question was addressed to me in quiz.

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

Answer 1

Right option is (b) \(\frac{3 sin⁡ a^3 – 2 sin a^2}{a}\)

The explanation is: Applying the Leibniz rule equation given by

\( f’ (α) = \int_p^q \frac{∂}{∂α} f (x,α) dx + f (q, α) \frac{dq}{dα} – f (p, α) \frac{dp}{dα} …..(1)\)

\(f (x,a) = \frac{sin⁡ax}{x}, p=a, q=a^2\) & further obtaining

\(f(q,a) = f(a^2,a) = \frac{sin⁡ a^3}{a^2}, \frac{dq}{da} = 2a\)

\(f(p,a) = f(a,a) = \frac{sin⁡ a^2}{a}, \frac{dp}{da} = 1\)

substituting all these values in (1) we get

\(f’(a) = \int_a^{a^2} \frac{∂}{∂α} (\frac{sin⁡ax}{x})dx + \frac{sin⁡ a^3}{a^2}.2a – \frac{sin⁡ a^2}{a}.1\)

\(\int_a^{a^2} \frac{1}{x} (cos(ax))(x)+ \frac{2 sin a^3 – sin^2 a}{a}\)

\([\frac{sin⁡ax}{x}]_a^{a^2} + \frac{2 sin a^3 – sin^2 a}{a} = \frac{sin⁡ a^3}{a} – \frac{sin⁡ a^2}{a} + \frac{2 sin a^3 – sin^2 a}{a}\)

thus \(f’(a) = \frac{3 sin⁡ a^3 – 2 sin a^2}{a}.\)