カテゴリー: 微分積分

\[ \int_{a}^{x}\int_{a}^{y_{1}}\cdots\int_{a}^{y_{n-1}}f\left(y_{n}\right)dy_{n}\cdots dy_{1}=\frac{1}{\left(n-1\right)!}\int_{a}^{x}\left(x-t\right)^{n-1}f\left(t\right)dt \]

\[ \int_{0}^{\frac{\pi}{2}}f\left(\cos x\right)dx=\int_{0}^{\frac{\pi}{2}}f\left(\sin x\right)dx \]

\[ \int_{-c}^{c}\frac{f_{e}\left(x\right)}{1+a^{x}}dx=\int_{0}^{c}f_{e}\left(x\right)dx \]

\[ \Arg\left(\alpha\right)\ne\pi,0<b\Rightarrow\int_{0}^{\infty}f\left(x,\alpha x^{b}\right)dx=\frac{1}{\alpha^{\frac{1}{b}}b}\lim_{R\rightarrow\infty}\int_{0}^{Re^{i\Arg\left(\alpha\right)}}f\left(\frac{t^{\frac{1}{b}}}{\alpha^{\frac{1}{b}}},t\right)t^{\frac{1}{b}-1}dt \]

\[ \int\log\left(x\right)f\left(x\right)dx=\left[\frac{d}{dt}\int x^{t}f\left(x\right)dx\right]_{t=0} \]

\[ \int z^{\alpha}e^{\beta z}dz=\frac{z^{\alpha}}{\beta\left(-\beta z\right)^{\alpha}}\Gamma\left(\alpha+1,-\beta z\right)+C \]

\[ f\left(x\right)=\int_{f^{\bullet}\left(a\right)}^{x}f'\left(x\right)dx-a \]

\[ \left(fg\right)^{(n)}=\sum_{k=0}^{n}C(n,k)f^{(k)}g^{(n-k)} \]

\[ \left(a^{x}\right)'=a^{x}\log a \]

\[ \frac{df(g(x))}{dx}=f'(g(x))g'(x) \]

\[ \left(f(x)g(x)\right)'=f'(x)g(x)+f(x)g'(x) \]

\[ \frac{df(x)}{dx}=\lim_{\Delta x\rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x} \]

\[ \int f'(g(x))g'(x)dx=f(g(x)) \]

\[ \int f\left(\sqrt{a^{2}-x^{2}}\right)dx=a\int f\left(a\cos t\right)\cos tdt\cnd{x=a\sin t} \]

\[ \int f(x)g(x)dx=\sum_{k=0}^{n-1}\left(-1\right)^{k}f^{(-(k+1))}(x)g^{(k)}(x)+(-1)^{n}\int f^{(-n)}(x)g^{(n)}(x)dx \]

\[ \frac{df^{\bullet}(x)}{dx}=\left(\frac{df(f^{\bullet}(x))}{df^{\bullet}(x)}\right)^{-1} \]