# Obliczyć całki.
## 1
$$\int x^{2}(1-x)dx=\int x^{2}-x^{3}dx=\int x^{2}dx+\int-x^{3}dx=\frac{x^{3}}{3}+\frac{-x^{4}}{4}+C$$
## 2 
$$\int x^{-2}dx=-x^{-1}+C$$
## 3
$$\int \frac{dx}{10x}=\ln|10x|+C$$
## 4
$$\int \frac{dx}{3x^{4}}=\ln|3x^{4}|+C$$
## 5
$$\int\left(\frac{3}{x^{2}}+\frac{4}{x^{3}}\right)dx=\int\frac{3}{x^{2}}dx+\int\frac{4}{x^{3}}dx=3\ln|x^{2}|+4\ln|x^{3}|+C$$
## 6
$$\begin{gathered}

\int\frac{(3-x)^{2}}{x^{3}}dx=\int\frac{9-6x+x^{2}}{x^{3}}dx=\int\frac{9}{x^{3}}dx+\int\frac{-6x}{x^3}dx+\int\frac{x^{2}}{x^{3}}dx\\
=
9\ln|x^{3}|-6\ln|x^2|+\ln|x|+C
\end{gathered}
$$
## 7
$$\int\frac{1-x^{2}}{x^{3}}dx=\int\frac{1}{x^{3}}dx+\int-\frac{x^{2}}{x^{3}}dx=\ln|x^{3}|-\ln|x|+C$$
## 8.
$$\begin{gather}
\int \frac{x^{4}-4}{x^{2}+2}dx=\int \frac{(x^{2}+2)(x^{2}-2)}{x^{2}+2}dx=\int x^{2}-2dx=\frac{x^{3}}{3}-2x+C
\end{gather}$$

## 9.
$$\int x\sqrt{x}\ dx=
\int x^{\frac{3}{2}}dx= \frac{x^{\frac{5}{2}}}{\frac{5}{2}}+c=\frac{2x^{\frac{5}{2}}}{5}+c$$
## 10.
$$
\int x\sqrt[4]{x^{3}}dx = \int\sqrt[4]{x^{7}}dx=\int x^{\frac{7}{4}}dx=\frac{x^{\frac{11}{4}}}{\frac{11}{4}}+C=\frac{4x^{\frac{11}{4}}}{11}+C=\frac{8\sqrt[4]{x^{3}}}{11}+C
$$
## 11.
$$
\begin{gather}
\int \sqrt{x}(\frac{1}{x}+x)dx=\int \frac{\sqrt{x}}{x}+x\sqrt{x}\ dx=\int x^{-\frac{1}{2}}+x^\frac{3}{2}dx=2x^\frac{1}{2}+\frac{2x^{\frac{5}{2}}}{5}+C\\
= \frac{10x^\frac{1}{2}+2x^{\frac{5}{2}}}{5}+C
\end{gather}
$$
## 12.
$$
\begin{gather}
\int \sqrt{x\sqrt{x}}\ dx=\int (x^{\frac{3}{2}})^{\frac{1}{2}}dx=\int x^{\frac{3}{4}}dx=\frac{4x^{\frac{7}{4}}}{7}+C
\end{gather}

$$
## 13.
$$
\begin{gather}
\int \frac{1}{\sqrt[3]{x^{2}}}dx=\int\sqrt[3]{x^{2}}^{-1}dx=\int ((x^{2})^{\frac{1}{3}})^{-1}dx=\int x^{-\frac{2}{3}}dx=3x^{\frac{1}{3}}+C
\end{gather}
$$
## 14. 
$$
\begin{gather}

\int\frac{x^{2}-x-2}{\sqrt[3]{x^{2}}}dx=\int\frac{x^{2}-x-2}{x^{\frac{2}{3}}}dx=\int \frac{x^{2}}{x^{\frac{2}{3}}}dx-\int \frac{x}{x^{\frac{2}{3}}}dx-\int \frac{2}{x^{\frac{2}{3}}}dx=\\
=\int x^{\frac{4}{3}}dx- \int x^{\frac{1}{3}}dx - \int \frac{2}{x^{\frac{2}{3}}}dx= \frac{3x^{\frac{7}{3}}}{7}- \frac{3x^{\frac{4}{3}}}{4}-2\cdot3x^\frac{1}{3}+C=\\
=\frac{3x^{2}\sqrt[3]{x}}{7} - \frac{3x\sqrt[3]{x}}{4} -6 \sqrt[3]{x}+C
 
\end{gather}
$$

## 15.
$$\begin{gather}
\int \frac{(x+\sqrt{x})(\sqrt{x}+\sqrt[4]{x})(\sqrt{x}-\sqrt[4]{x})}{x}dx=\int \frac{(x+\sqrt{x})(x - \sqrt{x})}{x}dx= \\
= \int\frac{x^{2}-x}{x}dx=\int x-1dx = \frac{x^{2}}{2}-x+C
\end{gather}$$
## 16.
$$
\int\frac{\sqrt{x+2\sqrt{x}+1}}{x}dx=\int \frac{x^{\frac{1}{2}}+1}{x}dx= \int \frac{x^{\frac{1}{2}}}{x}+\int \frac{1}{x}dx=2x^{\frac{1}{2}}+\ln|x|+C
$$
## 17.
$$\int \frac{\sqrt[3]{x}-1}{\sqrt[6]{x}-1}dx=\int\frac{x^{\frac{1}{3}}-1}{x^{\frac{1}{6}}-1}dx=\int\frac{x^{\frac{1}{3}}}{x^{\frac{1}{6}}-1}-\int\frac{1}{x^{\frac{1}{6}}-1}=$$
## 18.
$$\int (4^{x}+2^{-x})dx=\frac{4x}{\ln 4}+ \int (\frac{1}{2})^{x}dx=\frac{4^{x}}{\ln 4}+\frac{(\frac{1}{2})^{x}}{\ln \frac{1}{2}}+c$$
## 19.
$$
\begin{gathered}
\int5^{x}e^{x}dx= \int(5e)^x

\end{gathered}
$$