Integration of Weird Functions - 2

Calculus Level 5

Define f ( x ) f(x) on [ 0 , 1 ] [0,1] by f ( 0 ) = 0 f(0) = 0 and f ( x ) = lim ⁡ n → ∞ ( 1 + sin ⁡ ( π x ) ) n − 1 ( 1 + sin ⁡ ( π x ) ) n + 1 , x ∈ ( 0 , 1 ] \large f(x) = \lim_{n\to\infty} \frac{(1+\sin(\frac{\pi}{x}))^{n} -1}{(1+\sin(\frac{\pi}{x}))^{n} +1} , x \in (0,1]

Compute ∫ 0 1 f ( x ) d x \large \int_{0}^{1} f(x) \,dx If you think the function is un-integrable then input 1729 1729 as your answer


The answer is -0.386294361.

Arghyadeep Chatterjee by Arghyadeep Chatterjee , 22, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

we have f ( x ) = 0 f(x) = 0 for x = 1 , 1 2 , 1 3 , . . . . . x=1,\frac{1}{2},\frac{1}{3},.....

f ( x ) = − 1 f(x) = -1 for 1 2 < x < 1 \frac{1}{2}<x<1

f ( x ) = 1 f(x) = 1 for 1 3 < x < 1 2 \frac{1}{3}<x<\frac{1}{2}

f ( x ) = − 1 f(x) = -1 for 1 4 < x < 1 3 \frac{1}{4}<x<\frac{1}{3}

and so on..............

The derived set of points of discontinuity of f is finite. Hence the function is Riemann Integrable .

So we can represent the integral of this piecewise continuous function as an infinite sum .

∫ 0 1 f ( x ) d x = − 1 ( 1 − 1 2 ) + 1 ( 1 2 − 1 3 ) − 1 ( 1 3 − 1 4 ) . . . . . . . . \int_{0}^{1} f(x) dx = -1(1-\frac{1}{2}) + 1(\frac{1}{2} - \frac{1}{3}) -1(\frac{1}{3} - \frac{1}{4}) ........

= − ( 1 − 1 2 + 1 3 + . . . . . . . ) + ( 1 2 − 1 3 + 1 4 − . . . . . . . ) = -(1-\frac{1}{2} +\frac{1}{3}+.......) + (\frac{1}{2} - \frac{1}{3} + \frac{1}{4} -.......)

Using the fact that 1 − 1 2 + 1 3 − 1 4 + . . . . . = ln ⁡ ( 2 ) 1-\frac{1}{2} + \frac{1}{3} -\frac{1}{4} +..... = \ln(2)

We have our answer as 1 − 2 ln ⁡ ( 2 ) 1 - 2\ln(2)

PS:- I am not showing the evaluation of the limits as they are very elementary .

2 Helpful 0 Interesting 6 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...