# Function as an integral of itself

$\large g(x) = e^x \int_0^1 e^x g(x) \, dx$

Find $$g(0)$$.

Note by D K
2 years, 1 month ago

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$\large g(x)=e^x \underbrace{\displaystyle\int_0^1e^xg(x)\mathrm{d}x}_{\text{Constant=C}}=Ce^x$

$C=\displaystyle\int_0^1e^x(Ce^x)\mathrm{d}x=\frac C2(e^2-1)\implies C=0$

Hence $$\large g(x)=0\forall x\in \mathbb R$$

Hence $$g(0)=0$$.

- 2 years, 1 month ago

Explain, how is that a constant?

- 2 years, 1 month ago

Any definite integral with constant bounds is a constant.

- 2 years, 1 month ago

Couldn't $$g(x) = 0?$$ Just seems like $$g(0) = 0$$ then.

- 2 years, 1 month ago

g(0)=0

- 2 years, 1 month ago