The area bounded by the curve y = f(x),the co | vedclass

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(D) The area $A$ bounded by the curve $y = f(x)$,the $x$-axis,and the lines $x = 0$ and $x = x_1$ is given by the integral: $\int_{0}^{x_1} f(x) \, dx

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$x e^x + e^x$

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(D) The area $A$ bounded by the curve $y = f(x)$,the $x$-axis,and the lines $x = 0$ and $x = x_1$ is given by the integral: $\int_{0}^{x_1} f(x) \, dx = x_1 e^{x_1}$ To find $f(x)$,we differentiate both sides with respect to $x_1$ using the Fundamental Theorem of Calculus: $\frac{d}{dx_1} \left( \int_{0}^{x_1} f(x) \, dx \right) = \frac{d}{dx_1} (x_1 e^{x_1})$ $f(x_1) = \frac{d}{dx_1} (x_1) \cdot e^{x_1} + x_1 \cdot \frac{d}{dx_1} (e^{x_1})$ $f(x_1) = 1 \cdot e^{x_1} + x_1 e^{x_1}$ Replacing $x_1$ with $x$,we get: $f(x) = e^x + x e^x$

The area bounded by the curve $y = f(x)$,the coordinate axes,and the line $x = x_1$ is given by $x_1 \cdot e^{x_1}$. Therefore,$f(x)$ equals:

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