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${\displaystyle\int}\mathrm{d}x={\displaystyle\int}1\,\mathrm{d}x=x+c$,
for $x\in R$.
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${\displaystyle\int} x^a\,\mathrm{d}x=\frac{x^{a+1}}{a+1}+c$,
for $a\ne-1$, $x\in R-\{0\}$.
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${\displaystyle\int}\frac{\,\mathrm{d}x}{x}=\ln{|x|}+c$,
for $x\in R-\{0\}$.
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${\displaystyle\int}\frac{f'(x)}{f(x)}\,\mathrm{d}x=\ln{|f(x)|}+c$,
for $f(x)\ne0$, $x\in D(f)$.
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${\displaystyle\int}\mathrm{e}^{ax}\,\mathrm{d}x=\frac{\mathrm{e}^{ax}}{a}+c$,
for $a\ne0$, $x\in R$.
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${\displaystyle\int} a^x\,\mathrm{d}x=\frac{a^x}{\ln{a}}+c$,
for $a\geq0$, $a\ne1$, $x\in R$.
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${\displaystyle\int}\sin{ax}\,\mathrm{d}x=-\frac{\cos{ax}}{a}+c$,
for $a\ne0$, $x\in R$.
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${\displaystyle\int}\cos{ax}\,\mathrm{d}x=\frac{\sin{ax}}{a}+c$,
for $a\ne0$, $x\in R$.
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${\displaystyle\int}\frac{\,\mathrm{d}x}{\sin^2{ax}}=-\frac{\mathrm{cotg\,}{ax}}{a}+c$,
for $a\ne0$, $x\in R$, $x\ne\frac{k\pi}{a}$, $k\in Z$.
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${\displaystyle\int}\frac{\,\mathrm{d}x}{\cos^2{ax}}=\frac{\mathrm{tg\,}{ax}}{a}+c$,
for $a\ne0$, $x\in R$, $x\ne\frac{(2k+1)\pi}{2a}$, $k\in Z$.
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${\displaystyle\int}\sinh{ax}\,\mathrm{d}x=\frac{\cosh{ax}}{a}+c$,
for $a\ne0$, $x\in R$.
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${\displaystyle\int}\cosh{ax}\,\mathrm{d}x=\frac{\sinh{ax}}{a}+c$,
for $a\ne0$, $x\in R$.
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${\displaystyle\int}\frac{\,\mathrm{d}x}{\sinh^2{ax}}=-\frac{\mathrm{cotgh\,}{ax}}{a}+c$,
for $a\ne0$, $x\in R-\{0\}$.
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${\displaystyle\int}\frac{\mathrm{d}x}{\cosh^2{ax}}=\frac{\mathrm{tgh\,}{ax}}{a}+c$,
for $a\ne0$, $x\in R$.
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${\displaystyle\int}\frac{\mathrm{d}x}{x^2+a^2}=\frac1a\mathrm{arctg\,}{\frac{x}{a}}+c_1
=-\frac1a\mathrm{arccotg\,}{\frac{x}{a}}+c_2$,
for $a\ne0$, $x\in R$.
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${\displaystyle\int}\frac{\mathrm{d}x}{x^2-a^2}
={\displaystyle\int}\frac{1}{2a}\left[\frac{1}{x-a}-\frac{1}{x+a}\right]\,\mathrm{d}x
=\frac{1}{2a}\ln{\bigg|\frac{x-a}{x+a}\bigg|}+c$,
for $a\ne0$, $x\in R-\{\pm a\}$.
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${\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{a^2-x^2}}=\arcsin{\frac{x}{|a|}}+c_1
=-\arccos{\frac{x}{|a|}}+c_2$,
for $a\ne0$, $x\in\big(-|a|;\,|a|\big)$.
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${\displaystyle\int}\sqrt{a^2- x^2}\,\mathrm{d}x
=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{a^2-x^2}}$,
for $a\ne0$, $x\in\big(-|a|;\,|a|\big)$.
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${\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x^2+a^2}}=\ln{\big(x+\sqrt{x^2+a^2}\big)}+c$
for $a\ne0$, $x\in R$.
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${\displaystyle\int}\sqrt{x^2+ a^2}\,\mathrm{d}x
=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x^2+a^2}}$,
for $a\ne0$, $x\in R$.
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${\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x^2-a^2}}=\ln{\big|x+\sqrt{x^2-a^2}\big|}+c$,
for $x\in\big(-\infty;\,-|a|\big)\cup\big(|a|;\,\infty\big)$.
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${\displaystyle\int}\sqrt{x^2- a^2}\,\mathrm{d}x
=\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x^2-a^2}}$,
for $x\in\big(-\infty;\,-|a|\big)\cup\big(|a|;\,\infty\big)$.
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