1.0 KiB
1.0 KiB
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
$$\begin{gathered}
\int\frac{(3-x)^{2}}{x^{3}}dx=\int\frac{9-6x+x^{2}}{x^{3}}dx=\int\frac{9}{x^{3}}dx+\int\frac{-6x}{x^3}dx+\int\frac{x^{2}}{x^{3}}dx= \newline \newline 9\ln|x^{3}|-6\ln|x^2|+\ln|x|+C \end{gathered}
## 7
$$\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$$
## 10.
\int x\sqrt[4]{x^{3}}dx = \int\sqrt[4]{x^{7}}dx=\int x^{\frac{7}{4}}dx=
## 18.
$$\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$$