1 Calculus Review

1.1 Logarithms

\(\log_b a = x \leftrightarrow b^x = a\)

\(e^{c \ln x} = x^c\)

1.2 Derivative & Integration rules

Derivative	Integral
\(\frac{d}{dx} x^n = nx^{n-1}\)	\(\int x^n dx= \frac{x^{n+1}}{n+1}+c\)
\(\frac{d}{dx} n^x = n^x \ln x\)	\(\int n^x dx = \frac{n^x}{\ln n} + c\)
\(\frac{d}{dx} \ln x = \frac{1}{x}\)	\(\int \frac{1}{ax+b} dx = \frac{1}{a} \ln|ax+b| +c\)
\(\frac{d}{dx} e^x = e^x\)	\(\int e^x dx = e^x + c\)
\(\frac{d}{dx} \sin x = cos x\)	\(\int \sin x dx = -\cos x + c\)
\(\frac{d}{dx} \cos x = -\sin x\)	\(\int \cos x dx = \sin x + c\)
\(\frac{d}{dx} \tan x = \sec^2 x\)	\(\int \tan x = \ln|\sec x| + c\)

\(\int f(x) \pm g(x) dx = \int f(x) dx \pm \int g(x) dx\)

\(\int x f(x) = x F(x) + f(x)\)

1.2.1 Quotient Rule

\(\frac{d}{dx}(\frac{f( x )}{g( x )}) = \frac{f^\prime( x )g( x ) - f( x )g^\prime( x )}{g^2( x )}\)

1.2.2 Integration by substitution

\(u = g(x)\)

\(\int_a^b f(g(x)) g'(x) dx = \int_{g(a)}^{g(b)} f(u) du\)

1.2.3 Integration by parts

Assign \(u\) and \(dv\), differentiate \(u\) to find \(du\), integrate \(dv\) to find \(v\), then solve:

\(\int_a^b u dv = \left[uv\right]_a^b - \int_a^bvdu\)

1.3 Trigonometry

1.3.1 SOH CAH TOA

1.3.2 Basic Identities

1.3.3 Pythagorean Identities

\(\sin^2x + \cos^2x = 1\)

\(\tan^2x + 1 = \sec^2x\)

\(1 + \cot^2x = \csc^2x\)