Таблица на основни интеграли

Основни интеграли

• $\int x^{p}dx=\frac{x^{p+1}}{p+1}+C,p\ne -1$

• $\int \frac{dx}{x}=\ln |x|+C$

• $\int a^{x}dx=\frac{a^x}{\ln x}+C,a>0,a\ne 1$

• $\int e^{x}dx=e^x+C$

• $\int \sin xdx=-\cos x+C$

• $\int \cos xdx=\sin x+C$

• $\int \frac{dx}{{\cos }^{2}x}=\mathrm{tg}x+C$

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

• $\int \frac{dx}{\sqrt{1-x^2}}=\arcsin x+C$

• $\int \frac{dx}{1+x^2}=\mathrm{arctg}x+C$

Останати интеграли

• $\int \frac{dx}{ax+b}=\ln |ax+b|+C,a\ne 0$

• $\int \frac{dx}{x^2+a^2}=\frac{1}{a}\mathrm{arctg}\frac{x}{a}+C$

• $\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\ln \left|\frac{x-a}{x+a}\right|+C$

• $\int \frac{dx}{a{x}^2+bx+c}=\{\begin{array}{cc}\frac{1}{\sqrt{b^2-4ac}}\ln \left|\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right|+C,& b^2-4ac>0\\ \\ \frac{1}{\sqrt{4ac-b^2}}\mathrm{arctg}\frac{2ax+b}{4ac-b^2}+C,& b^2-4ac<0\end{array}$

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

• $\int \frac{dx}{\sqrt{a^2-x^2}}=\mathrm{sgn}a\cdot \arcsin \frac{x}{a}+C$

• $\int \frac{dx}{\sqrt{a{x}^2+bx+c}}=\frac{1}{\sqrt{a}}\ln |x+\frac{b}{2a}+\sqrt{x^2+\frac{b}{a}x+\frac{c}{a}}|+C$

• $\int \sqrt{x^2\pm a^2}dx=\frac{1}{2}[x\sqrt{x^2\pm a^2}+a^2\ln |x+\sqrt{x^2\pm a^2}|]+C$

• $\int \sqrt{a^2-x^2}dx=\frac{1}{2}[\mathrm{sgn}a\cdot a^2\arcsin \frac{x}{a}+x\sqrt{a^2-x^2}]+C$

• $\int \sqrt{a{x}^2+bx+c}dx=\frac{\sqrt{a}}{2}[(x+\frac{b}{2a})\sqrt{x^2+\frac{b}{a}x+\frac{c}{a}}+\frac{4ac-b^2}{4a^2}\ln |x+\frac{b}{2a}+\sqrt{x^2+\frac{b}{a}x+\frac{c}{a}}|]+C$