5. Grundformeln der Integration

$\begin{array}{l}1.\int x^{\mathrm{\alpha }}\mathit{dx}=\frac{x^{\mathrm{\alpha }+1}}{\mathrm{\alpha }+1}+C,\left(\mathrm{\alpha }\in R,\mathrm{\alpha }\ne -1\right)\\ 2.\int \frac{\mathit{dx}}{x}=\mathit{ln}\left|x\right|+C\\ 3.\int a^x\mathit{dx}=\frac{a^x}{\mathit{ln}a}+C,\left(a\in R,a>\mathrm{0,}a\ne 1\right)\\ 4.\int e^x\mathit{dx}=e^x+C\\ 5.\int \mathit{sin}\mathit{xdx}=-\mathit{cos}x+C\\ 6.\int \mathit{cos}\mathit{xdx}=\mathit{sin}x+C\\ 7.\int \frac{\mathit{dx}}{{\mathit{cos}}^{2}x}=\mathit{tan}x+C\\ 8.\int \frac{\mathit{dx}}{{\mathit{sin}}^{2}x}=-\mathit{cot}x+C\\ 9.\int \frac{\mathit{dx}}{\sqrt{a^2-x^2}}=\mathit{arcsin}\frac{x}{\mathrm{\alpha }}+C,\left(\mathrm{\alpha }\in R,\mathrm{\alpha }\ne 0\right)\\ 10.\int \frac{\mathit{dx}}{a^2+x^2}=\frac{1}{a}\mathit{arctan}\frac{x}{a}+C\\ 11.\int \frac{\mathit{dx}}{\sqrt{a^2\pm x^2}}=\mathit{ln}\left|x+\sqrt{x^2\pm a^2}\right|+C\\ 12.\int \frac{\mathit{dx}}{a^2-x^2}=\frac{1}{2a}\mathit{ln}\left|\frac{a+x}{a-x}\right|+C\\ 13.\int \sqrt{a^2-x^2}\mathit{dx}=\frac{1}{2}\left(x\sqrt{a^2-x^2}+a^2\mathit{arcsin}\frac{x}{a}\right)+C\\ 14.\int \sqrt{x^2+a^2}\mathit{dx}=\frac{1}{2}\left(x\sqrt{x^2+a^2}+a^2\mathit{ln}\left|x+\sqrt{x^2+a^2}\right|\right)+C\\ 15.\int \mathit{sinh}x\mathit{dx}=\mathit{cosh}x+C\\ 16.\int \mathit{cosh}x\mathit{dx}=\mathit{sinh}x+C\\ 17.\int \frac{\mathit{dx}}{{\mathit{cosh}}^{2}x}=\mathit{tanh}x+C\\ 18.\int \frac{\mathit{dx}}{{\mathit{sinh}}^{2}x}=-\mathit{coth}x+C\end{array}$