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AMiAL/ICT/Ćwiczenia/Zadania/Całki/Zadanie 1.md

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+# 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.
+$$
+\begin{gathered}
+\int5^{x}e^{x}dx= \int(5e)^x
+
+\end{gathered}
+$$