Формулы интегрирования


Основные формулы интегрирования


$$ \int d x=x+c $$
$$ \mathrm{k}(\mathrm{f}(\mathrm{x})) \mathrm{d} \mathrm{x}=\mathrm{k} \cdot \int \quad \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} $$
$$ \int(\mathrm{u}+\mathrm{v}+\mathrm{w}+\ldots) \mathrm{d} \mathrm{x}=\int \quad \mathrm{u} \mathrm{d} \mathrm{x}+\int_{.} \quad \mathrm{v} \mathrm{d} \mathrm{x}+\int_{.} \quad \mathrm{w} \mathrm{d} \mathrm{x}+\ldots $$
$$ \int \mathrm{uv}^{\prime} \mathrm{d} \mathrm{x}=\mathrm{uv}-\int \mathrm{vu}^{\prime} \mathrm{d} \mathrm{x} \quad \int \mathrm{u} \mathrm{d} \mathrm{v}=\mathrm{uv}-\int \mathrm{v} \mathrm{d} \mathrm{v} $$

Интегралы от рациональных функций (23 шт)


$$ \int x^{n} d x=\frac{x^{n+1}}{n+1}+c $$
$$ \int(a x+b)^{n} d x=\frac{(a x+b)^{n+1}}{a(n+1)}+c $$
$$ \int \frac{\mathrm{d} \mathrm{x}}{\mathrm{x}}=\ln |\mathrm{x}|+\mathrm{C} $$
$$ \int \frac{d x}{a x+b}=\frac{1}{a} \ln |a x+b|+c $$
$$ \int \frac{a x+b}{c x+d} d x=\frac{a}{c} x+\frac{b c-a d}{c^{2}} \ln |c x+d|+c $$
$$ \int \frac{d x}{(x+a) \cdot(x+b)}=\frac{1}{a-b} \ln \left|\frac{x+b}{x+a}\right|+c $$
$$ \int \frac{d x}{x^{2}-a^{2}}=\frac{1}{2 a} \ln \left|\frac{x-a}{x+a}\right|+c $$
$$ \int \frac{x d x}{(x+a) \cdot(x+b)}=\frac{1}{a-b}(a \cdot \ln |x+a|-b \cdot \ln |x+b|)+c $$
$$ \int \frac{x d x}{x^{2}-a^{2}}=\frac{1}{2} \ln \left|x^{2}-a^{2}\right|+c $$
$$ \int \frac{d x}{x^{2}+a^{2}}=\frac{1}{a} \operatorname{arctg}\left(\frac{x}{a}\right)+c $$
$$ \int \frac{x d x}{x^{2}+a^{2}}=\frac{1}{2} \ln \left|x^{2}+a^{2}\right|+c $$
$$ \int \frac{\mathrm{d} \mathrm{x}}{\left(\mathrm{x}^{2}+\mathrm{a}^{2}\right)^{2}}=\frac{1}{2 \mathrm{a}^{2}} \cdot \frac{\mathrm{x}}{\mathrm{x}^{2}+\mathrm{a}^{2}}+\frac{1}{2 \mathrm{a}^{3}} \operatorname{arctg}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{C} $$
$$ \int \frac{x d x}{\left(x^{2}+a^{2}\right)^{2}}=-\frac{1}{2} \cdot \frac{1}{x^{2}+a^{2}}+c $$
$$ \int \frac{x d x}{\left(x^{2}+a^{2}\right)^{3}}=-\frac{1}{4} \cdot \frac{1}{\left(x^{2}+a^{2}\right)^{2}}+C $$
$$ \begin{array}{c} \frac{d x}{a x^{2}+b x+c}=\frac{1}{\sqrt{b^{2}-4 a c}} \cdot \ln \left|\frac{2 a x+b-\sqrt{b^{2}-4 a c}}{2 a x+b+\sqrt{b^{2}-4 a c}}\right|+c \\ \left(b^{2}-4 a c>0\right) \end{array} $$
$$ \begin{array}{r} \int \frac{d x}{a x^{2}+b x+c}=\frac{2}{\sqrt{4 a c-b^{2}}} \cdot \operatorname{arctg}\left(\frac{2 a x+b}{\sqrt{4 a c-b^{2}}}\right)+c \\ \left(b^{2}-4 a c<0\right) \end{array} $$
$$ \int \frac{x d x}{a x^{2}+b x+c}=\frac{1}{2 a} \ln \left|a x^{2}+b x+c\right|-\frac{b}{2 a} \int \frac{d x}{a x^{2}+b x+c} $$
$$ \int \frac{x d x}{a x+b}=\frac{1}{a^{2}}(b+a x-b \cdot \ln |a x+b|)+c $$
$$ \int \frac{x^{2} d x}{a x+b}=\frac{1}{a^{3}}\left[\frac{1}{2}(a x+b)^{2}-2 b(a x+b)+b^{2} \ln |a x+b|\right]+c $$
$$ \int \frac{d x}{x(a x+b)}=\frac{1}{b} \ln \left|\frac{a x+b}{x}\right|+c $$
$$ \int \frac{d x}{x^{2}(a x+b)}=-\frac{1}{b x}+\frac{a}{b^{2}} \cdot \ln \left|\frac{a x+b}{x}\right|+c $$
$$ \int \frac{x d x}{(a x+b)^{2}}=\frac{1}{a^{2}}\left(\ln |a x+b|+\frac{b}{a x+b}\right)+c $$
$$ \int \frac{x^{2} d x}{(a x+b)^{2}}=\frac{1}{a^{3}}\left(b+a x-2 b \cdot \ln |a x+b|-\frac{b^{2}}{a x+b}\right)+c $$

Интегралы от трансцендентных функций (15 шт)


$$ \int e^{x} d x=e^{x}+c $$
$$ \int a^{x} d x=\frac{a^{x}}{\ln |a|}+C $$
$$ \int \frac{d x}{x \cdot \ln x}=\ln |\ln x|+c $$
$$ \int x^{n} \cdot \ln x d x=x^{n+1}\left[\frac{\ln x}{n+1}-\frac{1}{(n+1)^{2}}\right]+C $$
$$ \int e^{a x} \cdot \ln x d x=\frac{e^{a x} \cdot \ln x}{a}-\frac{1}{a} \int \frac{e^{a x}}{x} d x $$
$$ \int x^{n} \ln ^{m} x d x=\frac{x^{n+1}}{n+1} \cdot \ln ^{m} x-\frac{m}{n+1} \int x^{n} \ln ^{m-1} x d x $$
$$ \int \frac{x^{n}}{\ln ^{m} x} d x=-\frac{x^{n+1}}{(m-1) \cdot \ln ^{m-1} x}+\frac{n+1}{m-1} \int \frac{x^{n}}{\ln ^{m-1} x} d x $$
$$ \int \ln x d x=x \cdot \ln x-x+c $$
$$ \int \arcsin (x) d x=x \cdot \arcsin (x)+\sqrt{1-x^{2}}+c $$
$$ \int \operatorname{arctg}(x) d x=x \cdot \operatorname{arctg}(x)-\ln \sqrt{1+x^{2}}+c $$
$$ \int e^{a x} d x=\frac{e^{a x}}{a}+c $$
$$ \int x \cdot e^{a x} d x=\frac{e^{a x}}{a^{2}}(a x-1)+C $$
$$ \int \frac{a^{x}}{x^{n}} d x=-\frac{a^{x}}{(n-1) \cdot x^{n-1}}+\frac{\ln (a)}{n-1} \int \frac{a^{x}}{x^{n-1}} d x $$
$$ \int \operatorname{sh}(x) d x=\operatorname{ch}(x)+c \quad \int \operatorname{ch}(x) d x=\operatorname{sh}(x)+c $$

Интегралы от иррациональных функций (27 шт)


$$ \int \frac{\mathrm{d} \mathrm{x}}{\sqrt{\mathrm{ax}+\mathrm{b}}}=\frac{2}{\mathrm{a}} \cdot \sqrt{\mathrm{ax}+\mathrm{b}}+\mathrm{c} $$
$$ \int \sqrt{a x+b} d x=\frac{2}{3 a}(a x+b)^{1.5}+c $$
$$ \int \frac{x d x}{\sqrt{a x+b}}=\frac{2(a x-2 b)}{3 a^{2}} \cdot \sqrt{a x+b}+c $$
$$ \int x \sqrt{a x+b} d x=\frac{2(3 a x-2 b)}{15 a^{2}} \cdot(a x+b)^{1.5}+c $$
$$ \begin{array}{c} \int \frac{\mathrm{d} x}{(x+c) \cdot \sqrt{a x+b}}=\frac{1}{\sqrt{b-a c}} \cdot \ln \left|\frac{\sqrt{a x+b}-\sqrt{b-a c}}{\sqrt{a x+b}+\sqrt{b-a c}}\right|+c \\ (b-a c>0) \end{array} $$
$$ \begin{array}{r} \int \frac{d x}{(x+c) \cdot \sqrt{a x+b}}=\frac{1}{\sqrt{a c-b}} \cdot \operatorname{arctg}\left(\sqrt{\frac{a x+b}{a c-b}}\right)+c \\ (b-a c<0) \end{array} $$
$$ \int \sqrt{\frac{a x+b}{c x+d}} d x=\frac{1}{c} \cdot \sqrt{(a x+b) \cdot(c x+d)}-\frac{a d-b c}{c \cdot \sqrt{a c}} \cdot \operatorname{arctg}\left(\sqrt{\frac{a(c x+d)}{c(a x+b)}}\right)+c $$
$$ \int \frac{\mathrm{d} \mathrm{x}}{\mathrm{x} \cdot \sqrt{\mathrm{ax}+\mathrm{b}}}=\frac{1}{\sqrt{\mathrm{b}}} \cdot \ln \left|\frac{\sqrt{\mathrm{ax}+\mathrm{b}}-\sqrt{\mathrm{b}}}{\sqrt{\mathrm{ax}+\mathrm{b}}+\sqrt{\mathrm{b}}}\right|+\mathrm{c} $$
$$ \int \frac{d x}{x \cdot \sqrt{a x+b}}=\frac{2}{\sqrt{-b}} \cdot \operatorname{arctg}\left(\sqrt{\frac{a x+b}{-b}}\right)+c $$
$$ \int \frac{d x}{x^{2} \cdot \sqrt{a x+b}}=\frac{-\sqrt{a x+b}}{b x}-\frac{a}{2 b} \int \frac{d x}{x \cdot \sqrt{a x+b}} d x $$
$$ \int \frac{\sqrt{a x+b}}{x} d x=2 \cdot \sqrt{a x+b}+b \int \frac{d x}{x \cdot \sqrt{a x+b}} d x $$
$$ \int \sqrt{\frac{\mathrm{a}-\mathrm{x}}{\mathrm{b}+\mathrm{x}}} \mathrm{d} \mathrm{x}=\sqrt{(\mathrm{a}-\mathrm{x})(\mathrm{b}+\mathrm{x})}+(\mathrm{a}+\mathrm{b}) \arcsin \left(\sqrt{\frac{\mathrm{x}+\mathrm{b}}{\mathrm{a}+\mathrm{b}}}\right)+\mathrm{C} $$
$$ \int \sqrt{\frac{a+x}{b-x}} d x=-\sqrt{(a+x)(b-x)}-(a+b) \arcsin \left(\sqrt{\frac{b-x}{a+x}}\right)+c $$
$$ \int \frac{\mathrm{d} \mathrm{x}}{\sqrt{\mathrm{ax}^{2}+\mathrm{b} \mathrm{x}+\mathrm{c}}}=\frac{1}{\sqrt{\mathrm{a}}} \cdot \ln \left|2 \mathrm{ax}+\mathrm{b}+2 \sqrt{\left.\mathrm{a} \cdot ( a x^{2}+\mathrm{b} \mathrm{x}+\mathrm{c}\right)}\right|+\mathrm{C} $$
$$ \int \frac{d x}{\sqrt{a x^{2}+b x+c}}=-\frac{1}{\sqrt{a}} \cdot \arcsin \left(\frac{2 a x+b}{\sqrt{b^{2}-4 a c}}\right)+c $$
$$ \int \sqrt{a x^{2}+b x+c} d x=\frac{2 a x+b}{4 a} \cdot \sqrt{a x^{2}+b x+c}+\frac{4 a c-b^{2}}{8 a} \int \frac{d x}{\sqrt{a x^{2}+b x+c}} d x $$
$$ \int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \cdot \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \cdot \ln \left|x+\sqrt{x^{2}+a^{2}}\right|+c $$
$$ \int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \cdot \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \cdot \ln \left|x+\sqrt{x^{2}-a^{2}}\right|+c $$
$$ \int \frac{\mathrm{d} x}{\sqrt{x^{2}+a^{2}}}=\ln \left|x+\sqrt{x^{2}+a^{2}}\right|+c $$
$$ \int \frac{\mathrm{d} \mathrm{x}}{\sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}}}=\ln \left|\mathrm{x}+\sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}}\right|+\mathrm{C} $$
$$ \int \frac{x d x}{\sqrt{x^{2}+a^{2}}}=\sqrt{x^{2}+a^{2}}+c $$
$$ \int \frac{\sqrt{x^{2}-a^{2}}}{x} d x=\sqrt{x^{2}-a^{2}}+a \cdot \arcsin \left(\frac{a}{x}\right)+c $$
$$ \int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \cdot \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \cdot \arcsin \left(\frac{x}{a}\right)+c $$
$$ \int \frac{\sqrt{a^{2}-x^{2}}}{x} d x=\sqrt{a^{2}-x^{2}}+a \cdot \ln \left|\frac{x}{a+\sqrt{a^{2}-x^{2}}}\right|+c $$
$$ \int \frac{\mathrm{d} x}{\sqrt{\mathrm{a}^{2}-\mathrm{x}^{2}}}=\arcsin \left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{C} $$
$$ \int \frac{x d x}{\sqrt{a^{2}-x^{2}}}=-\sqrt{a^{2}-x^{2}}+c $$
$$ \int \frac{\mathrm{d} x}{x \cdot \sqrt{a^{2}-x^{2}}}=\frac{1}{a} \cdot \ln \left|\frac{x}{a+\sqrt{a^{2}-x^{2}}}\right|+c $$

Интегралы от тригонометрических функций (31 шт)


$$ \int \sin (x) d x=-\cos (x)+c \quad \int \cos (x) d x=\sin (x)+c $$ $$ \int \sin ^{2}(x) d x=\frac{x}{2}-\frac{1}{4} \sin (2 x)+c $$ $$ \int \cos ^{2}(x) d x=\frac{x}{2}+\frac{1}{4} \sin (2 x)+c $$ $$ \int \sin ^{n}(x) d x=-\frac{1}{n} \sin ^{n-1}(x) \cdot \cos (x)+\frac{n-1}{n} \cdot \int \sin ^{n-2}(x) d x $$ $$ \int \cos ^{n}(x) d x=\frac{1}{n} \cos ^{n-1}(x) \cdot \sin (x)+\frac{n-1}{n} \cdot \int \cos ^{n-2}(x) d x $$ $$ \int \frac{\mathrm{d} \mathrm{x}}{\sin (\mathrm{x})}=\ln \left|\mathrm{tg}\left(\frac{\mathrm{x}}{2}\right)\right|+\mathrm{C} \quad \int \frac{\mathrm{d} \mathrm{x}}{\cos (\mathrm{x})}=\ln \left|\operatorname{tg}\left(\frac{\mathrm{x}}{2}+\frac{\pi}{2}\right)\right|+\mathrm{C} $$ $$ \int \frac{d x}{\sin ^{2}(x)}=-\operatorname{ctg}(x)+c \quad \int \frac{d x}{\cos ^{2}(x)}=\operatorname{tg}(x)+c $$ $$ \int \sin (x) \cos (x) d x=-\frac{1}{4} \cos (2 x)+c $$ $$ \int \sin ^{2}(x) \cos (x) d x=\frac{1}{3} \sin ^{3}(x)+c $$ $$ \int \sin (x) \cos ^{2}(x) d x=-\frac{1}{3} \cos ^{3}(x)+c $$ $$ \int \sin ^{2}(x) \cos ^{2}(x) d x=\frac{1}{8} x-\frac{1}{32} \sin (4 x)+c $$ $$ \int \operatorname{tg}(x) d x=-\ln |\cos (x)|+c \quad \int \operatorname{ctg}(x) d x=\ln |\sin (x)|+c $$ $$ \int \frac{\sin (x)}{\cos ^{2}(x)} d x=\frac{1}{\cos (x)}+C \quad \int \frac{\sin ^{2}(x)}{\cos ^{2}(x)} d x=\operatorname{tg}(x)-x+C $$ $$ \int \frac{\sin ^{2}(x)}{\cos (x)} d x=\ln \left|\operatorname{tg}\left(\frac{x}{2}+\frac{\pi}{2}\right)\right|-\sin (x)+c $$ $$ \int \frac{\cos (x)}{\sin ^{2}(x)} d x=-\frac{1}{\sin (x)}+C \quad \int \frac{\cos ^{2}(x)}{\sin ^{2}(x)} d x=-\operatorname{ctg}(x)-x+C $$ $$ \int \frac{\cos ^{2}(x)}{\sin (x)} d x=\ln \left|\operatorname{tg}\left(\frac{x}{2}\right)\right|+\cos (x)+C $$ $$ \int \frac{d x}{\cos (x) \sin (x)}=\ln |\operatorname{tg}(x)|+c $$ $$ \int \frac{d x}{\sin ^{2}(x) \cos (x)}=-\frac{1}{\sin (x)}+\ln \left|\operatorname{tg}\left(\frac{x}{2}+\frac{\pi}{2}\right)\right|+C $$ $$ \int \frac{d x}{\sin (x) \cos ^{2}(x)}=\frac{1}{\cos (x)}+\ln \left|\operatorname{tg}\left(\frac{x}{2}\right)\right|+c $$ $$ \int \frac{d x}{\sin ^{2}(x) \cos ^{2}(x)}=\operatorname{tg}(x)-\operatorname{ctg}(x)+c $$ $$ \int \frac{\mathrm{d} \mathrm{x}}{\sin ^{n}(\mathrm{x})}=-\frac{1}{\mathrm{n}-1} \frac{\cos (\mathrm{x})}{\sin ^{\mathrm{n}-1}(\mathrm{x})}+\frac{\mathrm{n}-2}{\mathrm{n}-1} \cdot \int \frac{\mathrm{d} \mathrm{x}}{\sin ^{\mathrm{n}-2}(\mathrm{x})} \mathrm{d} \mathrm{x} $$ $$ \int \operatorname{tg}^{n}(x) d x=\frac{\operatorname{tg}^{n-1}(x)}{n-1}-\int \operatorname{tg}^{n-2}(x) d x $$ $$ \int \operatorname{ctg}^{n}(x) d x=-\frac{\operatorname{ctg}^{n-1}(x)}{n-1}-\int \operatorname{ctg}^{n-2}(x) d x $$ $$ \int \sin (x) \cos ^{n}(x) d x=-\frac{\cos ^{n+1}(x)}{n+1}+c $$ $$ \int \sin ^{n}(x) \cos (x) d x=\frac{\sin ^{n+1}(x)}{n+1}+c $$

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