8. Definite Integrals

Suppose the graph of a continuous function $f$ is shown below, and the area between the graph of $f$ and the $x$-axis from $x=0$ to $x=2$ is $3$ (above the $x$-axis), and from $x=2$ to $x=4$ is $2$ (below the $x$-axis). What is the value of the definite integral ${\int }_0^{4}f\left(x\right)dx$?

Evaluate the following integral:

${\int }_0^{3}4dx$

Let $g\left(x\right)={\int }_0^{x}f\left(t\right) dt$, where $f$ is the function whose graph is shown. Which of the following statements is true about $g^{\prime }\left(x\right)$?

Evaluate the following integral:

${\int }_{-1}^2(x^2-3x+2)dx$

Evaluate the following integral:

${\int }_1^4(2x+1)dx$

Given the definite integral $F\(\left\)(x\(\right\))=\(\int\)_{12}^{20x}\!\(\left\)(h^4+\(\frac{63h}{\sqrt{h^5}\)}\(\right\))\,dh$, find the derivative $F^{\(\prime\)}\(\left\)(x\(\right\))$.

Given the definite integral $F\left(x\right)={\int }_3^x[t^8-\sin \left(t^4\right)]dt$ , find the derivative $F^{\prime }\left(x\right)$.

Evaluate the following integral:

${\int }_1^2\frac{5}{x^2}dx$

Evaluate the following integral:

${\int }_0^1(2x^3-x^2+4x)dx$

Evaluate the following integral:

${\int }_2^3{x}^{\frac{5}{2}}dx$