${\displaystyle {\int }_a^{b}f{(x)}^{2}dx=}{\displaystyle {\int }_a^{b}f(x)dx}{\displaystyle {\int }_a^{b}f(x)dx}$

All continuous functions have an antiderivative.

$y=\text{ln}\left(x^2+1\right)$ over $\left[0,e\right]$

$y=\sqrt{x}+\frac{1}{x}$ over $\left[1,4\right]$

${\displaystyle {\int }_0^4\frac{3t}{\sqrt{1+6t^2}}dt}$

${\displaystyle {\int }_0^{\pi \text{/}4}{e}^{{\text{cos}}^{2}x}}\text{sin}x\text{cos}xdx$

${\displaystyle \int x\text{ln}\left(x^2\right)dx}$

${\displaystyle \int \frac{e^{2x}}{1+e^{4x}}dx}$

$\frac{d}{dx}{\displaystyle {\int }_1^{x^3}\sqrt{4-t^2}dt}$

$\frac{d}{dx}{\displaystyle {\int }_0^{\text{cos}x}{e}^{t^2}}dt$

The average cost per gigabyte of RAM can be approximated by the function $C\left(t\right)=8,500,000{\left(0.65\right)}^t,$ where $t$ is measured in years since 1980, and $C$ is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

The velocity of a bullet from a rifle can be approximated by $v\left(t\right)=6400t^2-6505t+2686,$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0\le t\le 0.5.$ What is the total distance the bullet travels in 0.5 sec?