### Теория:

Таблица основных интегралов:

 $\begin{array}{l}\phantom{\rule{0.147em}{0ex}}1\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int {x}^{\mathrm{\alpha }}\phantom{\rule{0.147em}{0ex}}\mathit{dx}=\frac{{x}^{\mathrm{\alpha }+1}}{\mathrm{\alpha }+1}+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }\ne -1\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}2\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{x}=\mathit{ln}\phantom{\rule{0.147em}{0ex}}\left|x\right|+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}3\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{1+{x}^{2}}=\mathit{arctg}\phantom{\rule{0.147em}{0ex}}x+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}4\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{\sqrt{1-{x}^{2}}}=\mathit{arcsin}\phantom{\rule{0.147em}{0ex}}x+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}5\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int {a}^{x}\mathit{dx}=\frac{{a}^{x}}{\mathit{ln}\phantom{\rule{0.147em}{0ex}}a}+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}6\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int {e}^{x}\mathit{dx}={e}^{x}+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}7\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \mathit{sin}\phantom{\rule{0.147em}{0ex}}x\phantom{\rule{0.147em}{0ex}}\mathit{dx}=-\mathit{cos}\phantom{\rule{0.147em}{0ex}}x+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}8\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \mathit{cos}\phantom{\rule{0.147em}{0ex}}x\phantom{\rule{0.147em}{0ex}}\mathit{dx}=\mathit{sin}\phantom{\rule{0.147em}{0ex}}x+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}9\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{\mathit{cos}}^{2}x}=\mathit{tg}\phantom{\rule{0.147em}{0ex}}x+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}10\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{\mathit{sin}}^{2}x}=-\mathit{ctg}\phantom{\rule{0.147em}{0ex}}x+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}11\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{x}^{2}-{a}^{2}}=\frac{1}{2a}\mathit{ln}\left|\frac{x-a}{x+a}\right|+C\phantom{\rule{0.147em}{0ex}}\left(a\ne 0\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}12\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{\sqrt{{x}^{2}+k}}=\mathit{ln}\left|x+\sqrt{{x}^{2}+k}\right|+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}13\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{x}^{2}+{a}^{2}}=\frac{1}{a}\mathit{arctg}\frac{x}{a}+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \\ \phantom{\rule{0.147em}{0ex}}14\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{\sqrt{{a}^{2}-{x}^{2}}}=\mathit{arcsin}\frac{x}{a}+C\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\end{array}$