Realiza la investigación encome Integrar: de resolver los ejercicios: 1. \( \int x^{3} \cdot e^{x} d x \) 2. \( \int|1-x| \cdot \sqrt{x} d x \) 3. \( \int \frac{3 x}{\sqrt[3]{x^{2}+3}} d x \)

8. If \( f(x)=x^{2}+10 \sin x \), show that there is a number \( c \) such that \( f(x)=1000 \).

integral \( \int_{0}^{1} \int_{e^{y}}^{e} \frac{x}{\ln x} d x d y \) (by

7. Find \( \lim _{x \rightarrow 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1} \) 8. If \( f(x)=x^{2}+10 \sin x \), show that there is a number \( c \) such that \( f(x)=1000 \)

USing first principle prove that \( y=\sin x \) then \( \frac{d y}{d x}=\cos x \)

11. Find the values of \( a \) and \( b \) that make \( f \) continuous everywhere. \[ f(x)=\left\{\begin{array}{ll}\frac{x^{2}-4}{x-2} & \text { if } x<2 \\ a x^{2}-b x+3 & \text { if } 2 \leq x<3 \\ 2 x-a+b & \text { if } x \geq 3\end{array}\right. \]