Question

Find a function f (x) and a number k such that
3 + ∫_k^x▒〖f(t)dt=e^5x 〗

Answer 1

Given function is $3 + \int_k^x f(t)dt = e^{5x}$

we can write it as $\int_k^x f(t)dt = e^{5x} - 3$

Differentiating both sides, we get

$f(x) = 5e^{5x}$

Now, from the equation given in question,

$3 + \int_k^x 5e^{5x}dt = e^{5x}$

We get

$3 + [e^{5x}] - e^{5k} = e^{5x} \implies k = \frac{1}{5}ln(3)$