MU DIVISION

Question 24

Definite Integrals★★★★Question Bank

Loading saved statistics and bookmark… You can read the question while this loads.

⏱ Time: 0:00

Question

Let

f(x)

and

g(x)

both be continuous, differentiable functions. Let

g(x)

be periodic with finite period

p

and let

[a,b]

be a closed finite interval on the real line. How many of the following statements are always true?

\mathrm{I.}\quad\int_a^b(f(x)-g(x))\,dx=\int_a^bf(x)\,dx-\int_a^bg(x)\,dx

\mathrm{II.}\quad\int_0^{np}g(x)\,dx=n\int_0^pg(x)\,dx

\mathrm{III.}\quad\lim_{n\to\infty}\int_a^bf(x)g(nx)\,dx=\frac1p\left(\int_a^bf(x)\,dx\right)\left(\int_0^pg(x)\,dx\right)

Loading practice…

Getting your questions ready.

Question

Let

f(x)

and

g(x)

both be continuous, differentiable functions. Let

g(x)

be periodic with finite period

p

and let

[a,b]

be a closed finite interval on the real line. How many of the following statements are always true?

\mathrm{I.}\quad\int_a^b(f(x)-g(x))\,dx=\int_a^bf(x)\,dx-\int_a^bg(x)\,dx

\mathrm{II.}\quad\int_0^{np}g(x)\,dx=n\int_0^pg(x)\,dx

\mathrm{III.}\quad\lim_{n\to\infty}\int_a^bf(x)g(nx)\,dx=\frac1p\left(\int_a^bf(x)\,dx\right)\left(\int_0^pg(x)\,dx\right)