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

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 \(g(0)=0\).

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Explain, how is that a constant?

Any definite integral with constant bounds is a constant.

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

g(0)=0

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## Comments

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

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

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

Hence \(g(0)=0\).

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Explain, how is that a constant?

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Any definite integral with constant bounds is a constant.

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

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g(0)=0

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