Основни интеграли ∫ x p d x = x p + 1 p + 1 + C , p ≠ − 1 {\displaystyle \int x^{p}\,dx={\frac {x^{p+1}}{p+1}}+C,\,\,\,\ p\neq -1} ∫ d x x = ln | x | + C {\displaystyle \int {\frac {dx}{x}}=\operatorname {ln} |x|+C} ∫ a x d x = a x ln x + C , a > 0 , a ≠ 1 {\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\operatorname {ln} x}}+C,\,\,\,\ a>0,\,\ a\neq 1} ∫ e x d x = e x + C {\displaystyle \int e^{x}\,dx=e^{x}+C} ∫ sin x d x = − cos x + C {\displaystyle \int \operatorname {sin} x\,dx=-\operatorname {cos} x+C} ∫ cos x d x = sin x + C {\displaystyle \int \operatorname {cos} x\,dx=\operatorname {sin} x+C} ∫ d x cos 2 x = tg x + C {\displaystyle \int {\frac {dx}{\operatorname {cos} ^{2}x}}=\operatorname {tg} x+C} ∫ d x sin 2 x = − ctg x + C {\displaystyle \int {\frac {dx}{\operatorname {sin} ^{2}x}}=-\operatorname {ctg} x+C} ∫ d x 1 − x 2 = arcsin x + C {\displaystyle \int {\frac {dx}{\sqrt {1-x^{2}}}}=\operatorname {arcsin} x+C} ∫ d x 1 + x 2 = arctg x + C {\displaystyle \int {\frac {dx}{1+x^{2}}}=\operatorname {arctg} x+C} Останати интеграли ∫ d x a x + b = ln | a x + b | + C , a ≠ 0 {\displaystyle \int {\frac {dx}{ax+b}}=\operatorname {ln} |ax+b|+C,\,\,\,\ a\neq 0} ∫ d x x 2 + a 2 = 1 a arctg x a + C {\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\operatorname {arctg} {\frac {x}{a}}+C} ∫ d x x 2 − a 2 = 1 2 a ln | x − a x + a | + C {\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}={\frac {1}{2a}}\operatorname {ln} \left|{\frac {x-a}{x+a}}\right|+C} ∫ d x a x 2 + b x + c = { 1 b 2 − 4 a c ln | 2 a x + b − b 2 − 4 a c 2 a x + b + b 2 − 4 a c | + C , b 2 − 4 a c > 0 1 4 a c − b 2 arctg 2 a x + b 4 a c − b 2 + C , b 2 − 4 a c < 0 {\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}}} ∫ d x x 2 ± a 2 = ln | x + x 2 ± a 2 | + C {\displaystyle \int {\frac {dx}{\sqrt {x^{2}\pm a^{2}}}}=\operatorname {ln} \left|x+{\sqrt {x^{2}\pm a^{2}}}\right|+C} ∫ d x a 2 − x 2 = sgn a ⋅ arcsin x a + C {\displaystyle \int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\operatorname {sgn} a\cdot \operatorname {arcsin} {\frac {x}{a}}+C} ∫ d x a x 2 + b x + c = 1 a ln | x + b 2 a + x 2 + b a x + c 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} ∫ x 2 ± a 2 d x = 1 2 [ x x 2 ± a 2 + a 2 ln | x + x 2 ± a 2 | ] + 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} ∫ a 2 − x 2 d x = 1 2 [ sgn a ⋅ a 2 arcsin x a + x a 2 − x 2 ] + 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} ∫ a x 2 + b x + c d x = a 2 [ ( x + b 2 a ) x 2 + b a x + c a + 4 a c − b 2 4 a 2 ln | x + b 2 a + x 2 + b a x + c a | ] + 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} Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.
![{\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}](//wikimedia.org/api/rest_v1/media/math/render/svg/b0f41da46a21697d5c7f42211bddb4a6069a2494)
![{\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}](//wikimedia.org/api/rest_v1/media/math/render/svg/e4d7d5f53fa0b0532929298f23db229a1021b17e)
![{\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}](//wikimedia.org/api/rest_v1/media/math/render/svg/f568c5176145c7ca52eb07793557511df3f15450)
![{\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}](http://wikimedia.org/api/rest_v1/media/math/render/svg/b0f41da46a21697d5c7f42211bddb4a6069a2494)
![{\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}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e4d7d5f53fa0b0532929298f23db229a1021b17e)
![{\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}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f568c5176145c7ca52eb07793557511df3f15450)