Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int\left\{x^{2}+e^{\log x}+\left(\frac{e}{2}\right)^{x}\right\} d x$

Solution:

Given:

$\int\left\{x^{2}+e^{\log x}+\left(\frac{e}{2}\right)^{x}\right\} d x$

By Splitting, we get,

$\Rightarrow \int x^{2} d x+\int e^{\log x} d x+\int\left(\frac{e}{2}\right)^{x} d x$

By applying formula,

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

$\Rightarrow \frac{x^{2+1}}{2+1}+\int e^{\log _{e} x} d x+\int\left(\frac{e}{2}\right)^{x} d x$

$\Rightarrow \frac{x^{3}}{3}+\int x d x+\frac{1}{\log \left(\frac{e}{2}\right)} \log \left(\frac{e}{2}\right)^{x}$

$\Rightarrow \frac{x^{3}}{3}+\int x d x+\frac{1}{\log \left(\frac{e}{2}\right)} \log \left(\frac{e}{2}\right)^{x}$

$\Rightarrow \frac{x^{3}}{3}+\frac{x^{2}}{2}+\frac{1}{\log \left(\frac{e}{2}\right)} \log \left(\frac{e}{2}\right)^{x}+c$

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