Differential and Integral Calculus Questions and Answers – Differentiation Under Integral Sign

«
»

This set of Differential and Integral Calculus Multiple Choice Questions & Answers (MCQs) focuses on “Differentiation Under Integral Sign”.

1. 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
View Answer

Answer: b
Explanation: 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).
advertisement

2. Which among the following correctly defines Leibnitz rule of a function given by \( f (α) = \int_a^b (x,α)dx\) where a & b are constants?
a) \(f’(α) = \frac{∂}{∂α}\int_a^b f (x,α) dx\)
b) \(f’(α) = \frac{d}{dα} \int_a^b f (α) dx\)
c) \(f’(α) = \int_a^b \frac{∂}{∂α} f (x,α) dx\)
d) \(f’(α) = \int_a^b \frac{d}{dα} f (x,α) dx\)
View Answer

Answer: c
Explanation: \(f’(α) = \int_a^b \frac{∂}{∂α} f (x,α) dx = \frac{d(f(α))}{dα} = \frac{d}{dα} \int_a^b f (x,α) dx.\)

3. Which among the following correctly defines Leibnitz rule of a function given by
\( f (α) = \int_a^b (x,α)dx\) where a & b are functions of α?
a) \(f’(α) = \int_a^b \frac{∂}{∂α} f(x,α) dx\)
b) \(f’(α) = \frac{d}{dα} \int_a^b f(x,α) dx\)
c) \(f’(α) = \int_a^b \frac{∂}{∂α} f (x,α) dx + f(b, α) \frac{da}{dα} – f(a, α) \frac{db}{dα}\)
d) \(f’(α) = \int_a^b \frac{∂}{∂α} f (x,α) dx + f(b, α) \frac{db}{dα} – f(a, α) \frac{da}{dα}\)
View Answer

Answer: d
Explanation: \(f’(α) = \int_a^b \frac{∂}{∂α} f (x,α) dx + f(b, α) \frac{db}{dα} – f(a, α) \frac{da}{dα}\) when a & b are constants
\(\frac{da}{dα} \& \frac{da}{dα} = 0\) which reduces the equation d into a

4. 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) \(\frac{3 sin a^3 – 3 sin^2 a}{6a}\)
View Answer

Answer: b
Explanation: 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 \(f’(a) = \frac{3 sin⁡ a^3 – 2 sin a^2}{a}.\)
advertisement

5. 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
View Answer

Answer: a
Explanation: \(\int_0^1 \frac{x^2-1}{log_2⁡x} dx\) 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).

Sanfoundry Global Education & Learning Series – Differential and Integral Calculus.

To practice all areas of Differential and Integral Calculus, here is complete set of 1000+ Multiple Choice Questions and Answers.

advertisement
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn