"Let $f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}}$ be a differentiable function for all x$\in$R. Then f(x) equals :"

Question

Accepted Answer

"$f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}}$ .... (1)

Differentiating both sides w.r.t. x

$f'(x) = {e^x}.f(x) + {e^x}$ (Using Newton L:eibnitz Theorem)

$\Rightarrow {{f'(x)} \over {f(x) + 1}} = {e^x}$

Integrating w.r.t. x

$\int {{{f'(x)} \over {f(x) + 1}}dx = \int {{e^x}dx} }$

$\Rightarrow \ln (f(x) + 1) = {e^x} + c$

Put x = 0

ln 2 = 1 + c ($\because$ f(0) = 1, from equation (1))

$\therefore$ $\ln (f(x) + 1) = {e^x} + \ln 2 - 1$

$\Rightarrow f(x) + 1 = 2.\,{e^{{e^x} - 1}}$

$\Rightarrow f(x) = 2{e^{{e^x} - 1}} - 1$"

Let $f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}}$ be a differentiable function for all x$\in$R. Then f(x) equals :

Solution

About this question

Drill 25 more like these. Every day. Free.

Let $f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}}$ be a differentiable function for all x$\in$R. Then f(x) equals :

Solution

About this question

More Definite Integration questions

Drill 25 more like these. Every day. Free.