Таблица на основни интеграли

Основни интеграли

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

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

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

• $\int e^{x}\,dx=e^{x} C$

• $\int \operatorname {sin} x\,dx=-\operatorname {cos} x C$

• $\int \operatorname {cos} x\,dx=\operatorname {sin} x C$

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

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

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

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

Останати интеграли

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

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

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

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

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

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

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

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

• $\int {\sqrt {a^{2}-x^{2}}}\,dx={\frac {1}{2}}\left[\operatorname {sgn} a\cdot a^{2}\operatorname {arcsin} {\frac {x}{a}} x{\sqrt {a^{2}-x^{2}}}\right] C$

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