JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

1 Hard

1 of 1

Application +4 | -1

For each positive integer

n

, define a function

f_n

[0,1]

as follows:

f_n(x)=\left\{\begin{array}{ccc} 0 & \text {if} & x=0 \\ \sin \frac{\pi}{2 n} & \text {if} & 0 < x \leq \frac{1}{n} \\ \sin \frac{2 \pi}{2 n} & \text {if} & \frac{1}{n} < x \leq \frac{2}{n} \\ \sin \frac{3 \pi}{2 n} & \text {if} & \frac{2}{n} < x \leq \frac{3}{n} \\ \sin \frac{n \pi}{2 n} & \text {if} & \frac{n-1}{n} < x \leq 1 \end{array}\right.

Then the value of

\lim \limits_{{n \rightarrow \infty}} \int\limits_{0}^1 f_n(x) d x

is:

View answer and explanation

Correct option: D

\frac{2}{\pi}

Option D (

\frac{2}{\pi}

) is correct because the limit of the integral is evaluated by recognizing it as a Riemann sum and converting it to a definite integral.

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