vault backup: 2023-05-26 08:45:39

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

@@ -56,9 +56,22 @@ $$
 \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$$

AMiAL/Ćwiczenia/Zadania/Całki/Zadanie 2.md

@@ -0,0 +1,3 @@
+# Obliczyć całki.
+## 1.
+$$\int x \cos x dx=\sin  x +C$$