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\begin{eqnarray} \int x^n\dx & = & \frac{x^{n+1}}{n+1} \pc,\qquad n\in\mathbb{R}-\{-1\} \\ \int \frac{1}{x}\dx & = & \ln|x| \pc \\ \int \ee^x\dx &= & \ee^x \pc \\ \int a^x\dx &= & \frac{a^x}{\ln a} \pc \\ \int \sin x\dx & = & -\cos x \pc \\ \int \cos x\dx & = & \sin x \pc \\ \int \frac{\dx}{\sin^2x} & = & -\cotg x \pc \\ \int \frac{\dx}{\cos^2x} & = & \tg x \pc \\ \int \frac{\dx}{x^2+a^2} & = & \frac{1}{a}\arctg\frac{x}{a} \pc \\ \int \frac{\dx}{x^2-a^2} & = & \frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| \pc \\ \int \frac{\dx}{\sqrt{a^2-x^2}} & = & \arcsin\frac{x}{a} \pc \\ \int \frac{\dx}{\sqrt{x^2+k}} & = & \ln \left| x+\sqrt{x^2+k}\right| \pc,\qquad k\neq0 \end{eqnarray}
\begin{eqnarray} \int f(x) \pm g(x) \dx & = & \int f(x) \dx \; \pm \; \int g(x) \dx \qquad \mbox{(integrace součtu funkcí)} \\ \int f(g(x))\cdot g'(x) \dx & = & \int f(t) \,\mathrm{d}t, \qquad t=g(x) \qquad \mbox{(substituční metoda)} \\ \int f(ax+b) \dx & = & \frac{1}{a}\int f(t) \,\mathrm{d}t, \qquad t=ax+b \qquad \mbox{(lineární substituce)} \\ \int f'(x)g(x) \dx & = & f(x)g(x) - \int f(x)g'(x) \dx \qquad \mbox{(metoda per partes)} \end{eqnarray}
\begin{eqnarray} \int 3x^2 + 2\cos x - \ee^x \dx & = & x^3 + 2\sin x - \ee^x \pc \\ \end{eqnarray} \begin{eqnarray} \int 2x\sin x^2 \dx & = & \left\{\begin{array}{c} \mathrm{subst.}\ t= x^2\\ \qquad\quad\mathrm{d}t=2x\,\dx\end{array} \right\} \; = \; \int \sin t\,\mathrm{d}t \\ \; & = & \; -\cos t \pc \; = \; -\cos x^2 \pc \nonumber \\ \end{eqnarray} \begin{eqnarray} \int \ln x\dx & = & \int 1\cdot\ln x \dx \;\;=\;\; \left\{\begin{array}{c} u'=1,\;\;\; u=x\\ v=\ln x,\; v'=1/x\end{array} \right\} \; = \; x\ln x - \int x\cdot \frac{1}{x}\dx \\ \; & = & \; x\ln x - x \pc \nonumber \\ \end{eqnarray} \begin{eqnarray} \int \frac{\ln(\arctg x)}{1+x^2}\dx & = & \left\{\begin{array}{c} \mathrm{subst.}\ t=\arctg x\\ \quad\mathrm{d}t=\frac{\dx}{1+x^2}\end{array} \right\} \; = \; \int \ln t\, \mathrm{d}t \; = \; t\ln t - t \pc \\ \; & = & \; (\arctg x)\ln(\arctg x) \; - \; \arctg x \pc \nonumber \\ \end{eqnarray}
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