vault backup: 2023-03-06 11:56:25

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

@@ -1,6 +1,26 @@
 # Obliczyć całki.
 ## 1
-$$\int x^{2}(1-x)dx=\int x^{2}-x^{3}dx=$$
+$$\int x^{2}(1-x)dx=\int x^{2}-x^{3}dx=\int x^{2}dx+\int-x^{3}dx=\frac{x^{3}}{3}+\frac{-x^{4}}{4}+C$$
+## 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
+$$\begin{gathered}
+
+\int\frac{(3-x)^{2}}{x^{3}}dx=\int\frac{9-6x+x^{2}}{x^{3}}dx=\int\frac{9}{x^{3}}dx+\int\frac{-6x}{x^3}dx+\int\frac{x^{2}}{x^{3}}dx=
+\newline
+\newline
+9\ln|x^{3}|-6\ln|x^2|+\ln|x|+C
+\end{gathered}
+$$
+## 7
+$$\int\frac{1-x^{2}}{x^{3}}dx=\int\frac{1}{x^{3}}dx+\int-\frac{x^{2}}{x^{3}}dx=\ln|x^{3}|-\ln|x|$$
+
 ## 9.
 $$\int x\sqrt{x}\ dx=
 \int x^{\frac{3}{2}}dx= \frac{x^{\frac{5}{2}}}{\frac{5}{2}}+c=\frac{2x^{\frac{5}{2}}}{5}+c$$