Please, I want to solve only the two paragraphs marked in red.Real Anylsis $417$

$2024-2$

Let $f(x)=x^3$ and $P$ any partiton of the interval $I=[0,1]$

Show that $f$ is Remann integrble,

$($Hint: recall that ${\displaystyle \underset{K=1}{\overset{n}{}}}{k}^3=(\frac{n(n+1)}{2}{)}^2,$ and $($:${\displaystyle \underset{K=1}{\overset{n}{}}}(k-1{)}^3=(\frac{n(n-1)}{2}{)}^2\}$

State whether the following improper integral is convergent or divergent:

${\displaystyle \underset{0}{\overset{}{}}}\frac{1}{x\sqrt[\phantom{2}]{x}}dx$

${\displaystyle \underset{0}{\overset{}{}}}\frac{x}{x^2+1}dx$

Evalute $F^{\text{'}}(x),$ for

$F(x)={\displaystyle \underset{x}{\overset{x^2}{}}}cos{t}^{2}dt$

$P(r)={\displaystyle \underset{0}{\overset{r^2}{}}}{e}^{t}dt$

$F(x)={\displaystyle \underset{4x}{\overset{x^3}{}}}ln(1+t)dt$

Prove that the function $f(x)=cxln(1+x)$ is Riemann integrable on $0,2$

$1$