積分の公式集
\int{x^n}dx = \frac{1}{n+1} x^{n+1} + C \quad \left(n \neq -1\right)
\int{\frac{1}{x}}dx = \ln{\left|x\right|} + C
\int{e^x}dx = e^{x} + C
\int{e^{ax}}dx = \frac{1}{a} e^{ax} + C
\int{a^x}dx = \frac{a^x}{\ln{a}} + C
\int{\sin{x}}dx = -\cos{x} + C
\int{\cos{x}}dx = \sin{x} + C
\int{\frac{1}{\cos^2{x}}}dx = \tan{x} + C
\int{\tan{x}}dx = -\ln{\left|\cos{x}\right|} + C
部分積分 (integration by parts)
\int f(x)g^{\prime}(x) = f(x)g(x) - \int f^{\prime}(x)g(x)dx
\int_a^b f(x)g^{\prime}(x) = \left[f(x)g(x)\right]_a^b - \int_a^b f^{\prime}(x)g(x)dx
置換積分 (integration by substitution)
x = g(t)とおくと
\int f(x)dx = \int f(g(t))\frac{dx}{dt}dt
x = g(t)とおくと
\int_a^b f(x)dx = \int_c^d f(g(t))\frac{dx}{dt}dt
ただし、a = g(c), b = g(d)