University:
Rice University
Course:
MATH 402 | Differential Geometry
Academic year:

2020
Views:

205

Pages:

4
Author:

Milagros W

Integral Calculus Formula Sheet Content: Derivative Rules: [<0>frac{d}{dx}(c)=0] [\frac{d}{dx}(x^{n})=nx^{n-1}] [<1>frac{d}{dx}(sin~x)=cos~x] [\frac{d}{dx}(cos~x)=-sin~x] [\frac{d}{dx}(tan~x)=sec^{2}x] [\frac{d}{dx}(cot~x)=-csc^{2}x] [\frac{d}{dx}(sec~x)=sec~x~tan~x] [\frac{d}{dx}(csc~x)=-csc~x~cot~x] [\frac{d}{dx}(a^{x})=a^{x}ln~a] [\frac{d}{dx}(e^{x})=e^{x}] [\frac{d}{dx}(cf(x))=c\frac{d}{dx}(f(x))] [\frac{d}{dx}(f(x)\pm g(x))=\frac{d}{dx}(f(x))\pm\frac{d}{dx}(g(x))] [\frac{d}{dx}(f(g(x)))=f'(g(x))g'(x)] [\frac{d}{dx}(f\cdot g)=f'g+fg'] [\frac{d}{dx}(\frac{f}{g})=\frac{fg'-fg'}{g^{2}}] Properties of Integrals: [\int kf(u)du=k\int f(u)du] [\int[f(u)\pm g(u)]du=\int f(u)du\pm\int g(u)du] [\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx] [\int_{-\infty}^{\infty}f(x)dx=0] [\int_{a}^{b}f^{\prime}(x)dx=f(b)-f(a)] [\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx] [f_{avg}=\frac{1}{b-a}\int_{a}^{b}f(x)dx] [\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx] if f(x) is even [\int_{-a}^{a}f(x)dx=0] if f(x) is odd [\int g(f(x))f'(x)dx=\int_{f(a)}^{f(b)}g(u)du] [\int udv=uv-\int vdu] Integration Rules: [\int du=u+C] [\int u^{n}du=\frac{u^{n+1}}{n+1}+C] [\int\frac{du}{u}=ln|u|+C] [\int e^{u}du=e^{u}+C] [\int a^{u}du=\frac{1}{ln~a}a^{u}+C] [\int sin~u~du=-cos~u+C] [\int cos~u~du=sin~u+C] [\int sec^{2}u~du=tan~u+C] [\int csc^{2}u~du=-cot~u+C] [\int sec~u~tan~u~du=sec~u+C] [\int csc~u~cot~u~du=-csc~u+C] [\int\frac{du}{a^{2}+u^{2}}=\frac{1}{a}arctan(\frac{u}{a})+C] [\int\frac{du}{\sqrt{a^{2}-u^{2}}}=arcsin(\frac{u}{a})+C] [\int\frac{du}{u\sqrt{u^{2}-a^{2}}}=\frac{1}{a}arcsec[\frac{|u|}{a}]+C]