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Таблица интегралов. Правила интегрирования

$$\int x^a dx=\frac{x^{a+1}}{a+1}+C, a \neq -1$$

$$\int dx=x+C$$

$$\int 0dx=C$$

$$\int \frac{dx}{x}=\ln|x|+C$$

$$\int \frac{dx}{1+x^{2}}=arctg(x)+C, a\neq 0$$

$$\int \frac{dx}{\sqrt{1-x^{2}}}=arcsin(x)+C, a>0$$

$$\int a^{x}dx= \frac{a^{x}}{\ln a}+C$$

$$\int e^{x}dx= e^{x}+C$$

$$\int \sin x dx= -\cos x+C$$

$$\int \cos x dx= \sin x+C$$

$$\int \frac{dx}{\cos^{2}x}= tg(x)+C$$

$$\int \frac{dx}{\sin^{2}x}= -ctg(x)+C$$

$$\int \frac{dx}{x^{2}-a^{2}}= \frac{1}{2a}\ln|\frac{x-a}{x+a}|+C, a\neq 0$$

$$\int \frac{dx}{\sqrt{x^{2}+k}}= \ln|x+\sqrt{x^{2}+k}|+C$$

$$\int \frac{dx}{x^{2}+a^{2}}= \frac{1}{a}arctg(\frac{x}{a})+C, a\neq 0$$

$$\int \frac{dx}{\sqrt{a^{2}-x^{2}}}= arcsin(\frac{x}{a})+C, a>0$$

$$\int sh (x) dx= ch(x)+C$$

$$\int ch (x) dx= sh(x)+C$$

$$\int \frac{dx}{ch^{2}x}= th(x)+C$$

$$\int \frac{dx}{sh^{2}x}= -cth(x)+C$$

Экспоненциальные функции

$$\int e^{cx} dx = \dfrac{1}{c}e^{cx} + C$$ $$\int a^{cx} dx = \dfrac{1}{c \ln(a)}a^{cx} + C, a > 0, a \neq 1$$ $$\int x\cdot e^{cx} dx = \dfrac{e^{cx}}{c^2}(cx - 1) + C$$ $$\int x^n\cdot e^{cx} dx = \dfrac{1}{c}x^n e^{cx} - \dfrac{n}{c} \int x^{n - 1}e^{cx} dx$$ $$\int \dfrac{e^{cx}}{x} dx = \ln|x| + \sum_{i = 1}^{\infty} \dfrac{(cx)^i}{i\cdot i!} + C$$ $$\int \dfrac{e^{cx}}{x^n} dx = \dfrac{1}{n - 1}\left (-\dfrac{e^{cx}}{x^{n - 1}} + \int \dfrac{e^{cx}dx}{x^{n - 1}}\right ), n \neq 1$$ $$\int e^{cx} \sin(bx) dx = \dfrac{e^{cx}}{c^2+b^2} (c\sin(bx)-b\cos(bx)) + C$$ $$\int e^{cx} \cos(bx) dx = \dfrac{e^{cx}}{c^2+b^2} (c\cos(bx)+b\sin(bx)) + C$$

Логарифмические функции

$$\int \ln(cx) dx = x \ln(cx) - x + C$$ $$\int \ln(cx)^n dx = x(ln(cx))^n-n\int(\ln(cx))^{n-1} dx$$ $$\int \dfrac{dx}{\ln(x)} = \ln|ln(x)| + \sum_{i = 1}^{\infty} \dfrac{(\ln x)^i}{i\cdot i!} + C$$ $$\int \dfrac{dx}{(\ln(x))^{n}} = -\dfrac{x}{(n - 1)(\ln(x))^{n - 1}} + \dfrac{1}{n - 1}\int \dfrac{dx}{(\ln(x))^{n - 1}}dx, n \neq 1$$ $$\int \dfrac{dx}{x(\ln(x))^n} dx = -\dfrac{1}{(n - 1)(\ln(x))^{n - 1}} + C, n \neq 1$$ $$\int \sin(\ln x)dx = \dfrac{x}{2}(\sin(\ln x) - cos(\ln x)) + C$$ $$\int \cos(\ln x)dx = \dfrac{x}{2}(\sin(\ln x) + cos(\ln x)) + C$$

Иррациональные функции

Пусть \(u(x) = \sqrt{x^2+a^2}\)

$$\int u(x) dx = \dfrac{1}{2}(x\cdot u(x)+a^2\ln(x+u(x))) + C$$ $$\int u(x)^3 dx = \dfrac{1}{4}x\cdot u^3(x) + \dfrac{1}{8}3a^2x\cdot u(x) + \dfrac{3}{8}a^4\ln\left (\dfrac{x+u(x)}{a}\right ) + C$$ $$\int x\cdot u^{2n+1}(x) dx = \dfrac{u^{2n+3}(x)}{2n+3} + C$$ $$\int \dfrac{u(x) dx}{x} = u(x)-a\ln|\dfrac{a+u(x)}{x}| + C$$ $$\int \dfrac{dx}{u(x)} = \ln(x+u(x)) + C$$ $$\int \dfrac{x dx}{u(x)} = u(x) + C$$ $$\int \dfrac{dx}{x u(x)} dx = -\dfrac{1}{a}\ln|\dfrac{a+u(x)}{x}| + C$$

Правила интегрирования

$$\Big(\int f(x)dx\Big)'=f(x)$$

$$d\Big(\int f(x)dx\Big)=f(x)dx$$

$$\int d(F(x))=F(x)+C$$

$$\int a f(x)dx= a\int f(x)dx, a - const$$

$$\int [f(x)\pm g(x)]dx= \int f(x)dx\pm \int g(x)dx$$

$$\int f(x)dx=F(x)+C, u=\varphi (x)\Rightarrow \int f(u)du=F(u)+C$$

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