Question

Evaluate the following definite integrals as limit of sums.
${\int }_0^4(x+e^{2x})dx$

Answer 1

$\begin{array}{c}{\int }_0^4\left(x+e^{2x}\right)dx\\ \Rightarrow {\int }_0^{4}xdx+{\int }_0^4{e}^{2x}dx\\ \Rightarrow \frac{1}{2}{\left[x^2\right]}_0^4+\frac{1}{2}{\left[e^{2x}\right]}_0^4+c\\ \Rightarrow \frac{1}{2}\times 16=\frac{1}{2}\left[e^8-1\right]+c\\ 8+\frac{1}{2}\left[e^8-1\right]+c.\end{array}$