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Таблиця інтегралів

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    60234

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    1. \(\int \frac{1}{x(a x+b)} d x=\frac{1}{b} \ln \left|\frac{x}{a x+b}\right|+C \)
    2. \(\int \frac{x}{a x+b} d x=\frac{x}{a}-\frac{b}{a^{2}} \ln |a x+b|+C \)
    3. \(\int \frac{1}{(x+a)(x+b)} d x=\frac{1}{b-a}(\ln |x+a|-\ln |x+b|)+C=\frac{1}{b-a} \ln \left|\frac{x+a}{x+b}\right|+C, \quad a \neq b \)
    4. \(\int \frac{1}{a^{2}-x^{2}} d x=\frac{1}{2 a} \ln \left|\frac{x+a}{x-a}\right|+C \)
    5. \(\int \frac{1}{\sqrt{x^{2} \pm a^{2}}} d x=\ln \left|x+\sqrt{x^{2} \pm a^{2}}\right|+C \)
    6. \(\int \sqrt{x^{2} \pm a^{2}} d x=\frac{x}{2} \sqrt{x^{2} \pm a^{2}}+\frac{a^{2}}{2} \ln \left|x+\sqrt{x^{2} \pm a^{2}}\right|+C \)
    7. \(\int x(a x+b)^{n} d x=\frac{(a x+b)^{n+1}}{a}\left(\frac{a x+b}{n+2}-\frac{b}{n+1}\right)+C, \quad n \neq-1,-2 \)
    8. \(\int x^{n} e^{a x} d x=\frac{1}{a} x^{n} e^{a x}-\frac{n}{a} \int x^{n-1} e^{a x} d x+C\)