Limit of a Definite Integral #2

Calculus Level 5

$\large f(x) = \frac2{e^x - e^{-x}} \left( 1 + \int_1^x f(t) \, dt \right)$

Suppose a function $f$ defined on $x>0$ satisfy the equation above, find the value of $\displaystyle \left \lfloor 100 \cdot \lim_{a\to\infty} \int_{\frac1a}^a f(t) \, dt \right \rfloor$ .

The answer is 216.

1 solution

Satyajit Mohanty
Jul 11, 2015

Good problem! My solution takes the derivative of both sides of the original functional equation where I obtained the ODE:

f'(x) + [[cosh(x) -1]/sinh(x)]*f(x) = 0; f(1) = csch(1)

which yields the function:

f(x) = [coth(1/2)]*[tanh(x/2) / sinh(x)].

Using this function in the following "floored" limit still gives me 216 for a final result!

tom engelsman - 5 years, 11 months ago

I did in the same way.

Shounak Ghosh - 5 years, 10 months ago

Same as i DID....(+1)

Rishabh Deep Singh - 5 years, 2 months ago

Limit of a Definite Integral #2

The answer is 216.

1 solution

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