Evaluate the following integrals:

Question:

Evaluate the following integrals:]

$\int\left(x^{e}+e^{x}+e^{e}\right) d x$

Solution:

Given:

$\int\left(x^{e}+e^{x}+e^{e}\right) d x$

By Splitting, we get,

$\Rightarrow \int x^{e} d x+\int e^{x} d x+\int e^{e} d x$

By using the formula,

$\int x^{n} d x=\frac{x^{n+1}}{n+1}$

$\Rightarrow \frac{x^{e+1}}{e+1}+\int e^{x} d x+\int e^{e} d x$

By applying the formula,

$\int a^{x} d x=\frac{a^{x}}{\log a}$

$\Rightarrow \frac{x^{e+1}}{e+1}+\frac{e^{x}}{\log _{e} e}+\int e^{e} d x$

We know that,

$\int \mathrm{kdx}=\mathrm{kx}+\mathrm{c}$

$\Rightarrow \frac{x^{e+1}}{e+1}+\frac{e^{x}}{\log _{e} e}+e^{e} x+c$

$\Rightarrow \frac{x^{e+1}}{e+1}+\frac{e^{x}}{\log _{e} e}+e^{e} x+c$

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