### Teorija

Zinot, ka ${x}^{2}=2x$, varam rakstīt, ka$\int 2\mathit{xdx}={x}^{2}+C$. Līdzīgā veidā, izmantojot

atvasināšanas pamatformulas, ir iegūtas integrēšanas pamatformulas. Tās ir šādas:

$\begin{array}{l}1.\phantom{\rule{0.147em}{0ex}}\int {x}^{\mathrm{\alpha }}\mathit{dx}=\frac{{x}^{\mathrm{\alpha }+1}}{\mathrm{\alpha }+1}+C,\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }\in R,\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\ne -1\right)\\ 2.\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{x}=\mathit{ln}\left|x\right|+C\\ 3.\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}}\left(\mathrm{\alpha }\in R,\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\ne -1\right)\\ 4.\phantom{\rule{0.147em}{0ex}}\int {e}^{x}\mathit{dx}={e}^{x}+C\\ 5.\phantom{\rule{0.147em}{0ex}}\int \mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathit{xdx}=-\mathit{cos}\phantom{\rule{0.147em}{0ex}}x+C\\ 6.\phantom{\rule{0.147em}{0ex}}\int \mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathit{xdx}=\mathit{sin}\phantom{\rule{0.147em}{0ex}}x+C\\ 7.\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{\mathit{cos}}^{2}x}=\mathit{tg}\phantom{\rule{0.147em}{0ex}}x+C\\ 8.\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{\mathit{sin}}^{2}x}=-\mathit{ctgx}+C\\ 9.\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{\sqrt{{a}^{2}-{x}^{2}}}=\mathit{arcsin}\frac{x}{\mathrm{\alpha }}+C,\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }\in R,\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\ne 0\right)\\ 10.\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{a}^{2}+{x}^{2}}=\frac{1}{a}\mathit{arctg}\frac{x}{a}+C\\ 11.\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{\sqrt{{a}^{2}±{x}^{2}}}=\mathit{ln}\left|x+\sqrt{{x}^{2}±{a}^{2}}\right|+C\\ 12.\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{a}^{2}-{x}^{2}}=\frac{1}{2a}\mathit{ln}\left|\frac{a+x}{a-x}\right|+C\\ 13.\phantom{\rule{0.147em}{0ex}}\int \sqrt{{a}^{2}-{x}^{2}}\mathit{dx}=\frac{1}{2}\left(x\sqrt{{a}^{2}-{x}^{2}}+{a}^{2}\mathit{arcsin}\frac{x}{a}\right)+C\\ 14.\phantom{\rule{0.147em}{0ex}}\int \sqrt{{x}^{2}+{a}^{2}}\mathit{dx}=\frac{1}{2}\left(x\sqrt{{x}^{2}+{a}^{2}}+{a}^{2}\mathit{ln}\left|x+\sqrt{{x}^{2}+{a}^{2}}\right|\right)+C\\ 15.\phantom{\rule{0.147em}{0ex}}\int \mathit{sh}\phantom{\rule{0.147em}{0ex}}\mathit{xdx}=\mathit{ch}\phantom{\rule{0.147em}{0ex}}x+C\\ 16.\phantom{\rule{0.147em}{0ex}}\int \mathit{ch}\phantom{\rule{0.147em}{0ex}}\mathit{xdx}=\mathit{sh}\phantom{\rule{0.147em}{0ex}}x+C\\ 17.\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{\mathit{ch}}^{2}x}=\mathit{th}\phantom{\rule{0.147em}{0ex}}x+C\\ 18.\phantom{\rule{0.147em}{0ex}}\int \frac{\mathit{dx}}{{\mathit{sh}}^{2}x}=-\mathit{cth}\phantom{\rule{0.147em}{0ex}}x+C\end{array}$