Chapter 8 积分 - 公式表

\[\int_a^b f(x) dx = F(b) - F(a)\]

\[\frac{d}{dx}\int_a^x f(t) dt = f(x)\]

\[\int_a^b [f(x) + g(x)] dx = \int_a^b f(x) dx + \int_a^b g(x) dx\] \[\int_a^b kf(x) dx = k\int_a^b f(x) dx\] \[\int_a^b f(x) dx = -\int_b^a f(x) dx\]

\[\text{面积} = \int_a^b f(x) dx\]

\[\text{面积} = \left|\int_a^b f(x) dx\right|\]

\[\text{面积} = \int_a^c |f(x)| dx + \int_c^b |f(x)| dx\]

\[\text{面积} = \int_a^b [f(x) - g(x)] dx\]

\[\text{面积} = \int_a^b |f(x) - g(x)| dx\]

\[\text{面积} = \text{几何图形面积} - \int_a^b f(x) dx\]

\[\int_a^b f(x) dx \approx \frac{h}{2}[y_0 + 2(y_1 + y_2 + \cdots + y_{n-1}) + y_n]\]

\[\text{梯形面积} = \frac{1}{2}(y_0 + y_1)h\]

\[\text{误差} \approx \frac{(b-a)^3}{12n^2} \max |f''(x)|\]

\[\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)\] \[\int x^{-1} dx = \ln|x| + C\]

\[\int e^x dx = e^x + C\] \[\int a^x dx = \frac{a^x}{\ln a} + C\] \[\int \frac{1}{x} dx = \ln|x| + C\]

\[\int \sin x dx = -\cos x + C\] \[\int \cos x dx = \sin x + C\] \[\int \sec^2 x dx = \tan x + C\]

\[\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx\] \[\int kf(x) dx = k\int f(x) dx\]