"MATEMĀTIKA 11. KLASEI"
Integrēšana ir atvasināšanai apgrieztā darbība. Izmantojot atvasināšanas pamatformulas, ir iegūtas integrēšanas pamatformulas.

Atvasināšanas pamatformulas
$\begin{array}{l}{x}^{\prime }=1\\ \phantom{\rule{0.147em}{0ex}}{\left({x}^{\mathrm{\alpha }}\right)}^{\prime }=\mathrm{\alpha }\cdot {x}^{\mathrm{\alpha }-1}\\ {\left(\mathrm{ln}\phantom{\rule{0.147em}{0ex}}x\right)}^{\prime }=\frac{1}{x}\\ {\left({e}^{x}\right)}^{\prime }={e}^{x}\\ {\left(\mathrm{sin}\phantom{\rule{0.147em}{0ex}}x\right)}^{\prime }=\mathrm{cos}\phantom{\rule{0.147em}{0ex}}x\\ {\left(\mathrm{cos}\phantom{\rule{0.147em}{0ex}}x\right)}^{\prime }=-\mathrm{sin}\phantom{\rule{0.147em}{0ex}}x\end{array}$

Integrēšanas pamatformulas

$\int \mathit{dx}=\int 1\cdot \mathit{dx}=x+C$

$\int {x}^{\mathrm{\alpha }}\mathit{dx}=\frac{{x}^{\mathrm{\alpha }+1}}{\mathrm{\alpha }+1}+C,\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }\in \mathrm{ℝ},\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\ne -1\right)$

$\int \frac{1}{x}\mathit{dx}=\mathrm{ln}\phantom{\rule{0.147em}{0ex}}\left|x\right|+C$

$\int {e}^{x}\mathit{dx}={e}^{x}+C$

$\begin{array}{l}\int \mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathit{xdx}=-\mathrm{cos}\phantom{\rule{0.147em}{0ex}}x+C\\ \int \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathit{xdx}=\mathrm{sin}\phantom{\rule{0.147em}{0ex}}x+C\end{array}$

Pārbaudīsim ar atvasināšanas darbību formulu $\int \frac{1}{x}\mathit{dx}=\mathrm{ln}\phantom{\rule{0.147em}{0ex}}\left|x\right|+C$.
Ja $$x>0$$, tad ${\left(\mathrm{ln}\phantom{\rule{0.147em}{0ex}}\left|x\right|\right)}^{\prime }={\left(\mathrm{ln}\phantom{\rule{0.147em}{0ex}}x\right)}^{\prime }=\frac{1}{x}$.
Ja $$x<0$$, tad ${\left(\mathrm{ln}\phantom{\rule{0.147em}{0ex}}\left|x\right|\right)}^{\prime }={\left(\mathrm{ln}\left(-x\right)\right)}^{\prime }=\frac{1}{-x}\cdot {\left(-x\right)}^{\prime }=\frac{1}{x}$.