Question

Evaluate ${\int }_1^3(x^2+3x+e^x)dx$ as the limit of the sum.

Answer 1

${\int }_1^3(x^3+3x+e^x)dx$

$={[\frac{x^4}{4}+\frac{3x^2}{2}+e^x]}_1^3$

$=[\frac{3^4}{4}+\frac{3\times 3^2}{2}+e^3]-[\frac{1^4}{4}+\frac{3\times 1^2}{2}+e^1]$

$=\frac{81}{4}+\frac{27}{2}+e^3-\frac{1}{4}-\frac{3}{2}-e$

$=\frac{81-1}{4}+\frac{27-3}{2}+e^3-e$

$=\frac{80}{4}+\frac{24}{2}+e^3-e$

$=20+12+e^3-e$

$=32+e^3-e$