Таблица на основни интеграли

Основни интеграли

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

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

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

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

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

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

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

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

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

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

Останати интеграли

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

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

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

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

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

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

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

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

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

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