Calc AB Final

d/dx[a^u]

a^{u}\ln\left(a\right)u^{\prime}

d/dx[log base a u]

\frac{u^{\prime}}{u\ln a}

\int_{}^{}a^{u}du

\left(\frac{1}{\ln a}\right)a^{u}+C

d/dx[arcsin u]

\frac{u^{\prime}}{\sqrt{1-u^2}}

d/dx[arctan u]

\frac{u^{\prime}}{1+u^2}

d/dx[arcsec u]

\frac{u^{\prime}}{\left|u\right|\sqrt{u^2-1}}

\int_{}^{}\frac{1}{\sqrt{a^2-u^2}}du

\arcsin\left(\frac{u}{a}\right)+\mathbb{C}

\int_{}^{}\frac{1}{a^2+u^2}du

\frac{1}{a}\arctan\left(\frac{u^{}}{a}\right)+C

\int_{}^{}\frac{1}{u\sqrt{u^2-a^2}}du

\frac{1}{a}\operatorname{arcsec}\left(\frac{\left|u\right|}{a}\right)+C

\int_{}^{}\sin xdx

-\cos x+C

\int_{}^{}\cos xdx

\sin x+C

\int_{}^{}\tan xdx

-\ln\left|\cos x\right|+C

\ln\left|\sin x\right|+C

\int_{}^{}\sec xdx

\ln\left|\sec x+\tan x\right|+C

\int_{}^{}\csc xdx

-\ln\left|\csc x+\cot x\right|+C

Compound Interest (n times)

A=P\left(1+\frac{r}{n}\right)^{nt}

Continuous Compounding

A=Pe^{rt}

Newton’s Law of Cooling

dy/dt=k\left(y-T_{surr}\right)

y=temp of object

t=time

Tsurr=temp of surrounding