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inta^(x)e^(x)dx.

$\int a^{x} e^{x} d x$ .

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Gaussian integral

Full solution

Q.

\int a^{x} e^{x} d x

.

Antiderivative of $e^{x}$ : The antiderivative of $e^{x}$ is $e^{x}$ itself, since the derivative of $e^{x}$ is $e^{x}$ . Therefore, we can write the antiderivative as $F(x) = e^{x} + C$ , where $C$ is the constant of integration.
Evaluation at Upper and Lower Limits: We will evaluate the antiderivative at the upper limit of integration, which is $x$ , and then subtract the evaluation of the antiderivative at the lower limit of integration, which is $a$ . This gives us $F(x) - F(a) = e^{x} - e^{a}$ .
Final Answer: The definite integral of $e^{x}$ from $a$ to $x$ is therefore $e^{x} - e^{a}$ . This is the final answer.

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