Calculus 2 Equation Sheet
Basic Trig Identities
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\[ \sin(x)=\frac{1}{\csc(x)} \]
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\[ \cos(x)=\frac{1}{\sec(x)} \]
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\[ \tan(x)=\frac{\sin(x)}{\cos(x)} \]
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\[ \cot(x)=\frac{1}{\tan(x)} \]
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\[ \sin(-x)=-\sin(x) \]
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\[ \cos(-x)=\cos(x) \]
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\[ \tan(-x)=-\tan(x) \]
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\[ \sin^2(x)+\cos^2(x)=1 \]
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\[ \tan^2(x)+1=\sec^2(x) \]
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\[ \cot^2(x)+1=\csc^2(x) \]
Power Reduction
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\[ \sin^2(\theta)=\frac{1}{2}(1-\cos(2\theta)) \]
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\[ \cos^2(\theta)=\frac{1}{2}(1+\cos(2\theta)) \]
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\[ \tan^2(\theta)=\frac{1-\cos(2\theta)}{1+\cos(2\theta)} \]
Trig Derivatives
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\[ \frac{d}{dx}\sin(x)=\cos(x) \]
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\[ \frac{d}{dx}\cos(x)=-\sin(x) \]
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\[ \frac{d}{dx}\tan(x)=\sec^2(x) \]
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\[ \frac{d}{dx}\cot(x)=-\csc^2(x) \]
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\[ \frac{d}{dx}\sec(x)=\sec(x)\tan(x) \]
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\[ \frac{d}{dx}\csc(x)=-\csc(x)\cot(x) \]
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\[ \frac{d}{dx}\arcsin(x)=\frac{1}{\sqrt{1-x^2}} \]
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\[ \frac{d}{dx}\arctan(x)=\frac{1}{1+x^2} \]
Trig Integrals
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\[ \cos(x)dx=\sin(x)+C \]
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\[ \sin(x)dx=-\cos(x)+C \]
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\[ \int\sec^2(x)dx=\tan(x)+C \]
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\[ \int\tan(x)dx=-\ln|\cos(x)|+C \]
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\[ \int\cot(x)dx=\ln|\sin(x)|+C \]
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\[ \int\sec(x)\tan(x)dx=\sec(x)+C \]
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\[ \int\csc^2(x)dx=-\cot(x)+C \]
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\[ \int\csc(x)\cot(x)dx=-\csc(x)+C \]
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\[ \int\frac{u'}{a^2+u^2}du=\frac{1}{a}\arctan\left(\frac{u}{a}\right)+C \]
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\[ \int\frac{u'}{\sqrt{a^2-u^2}}du=\arcsin\left(\frac{u}{a}\right)+C \]
Volume
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\[ V=\pi\int_a^br^2dx\tag{disc} \]
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\[ V=\pi\int_a^b(R^2-r^2)dx\tag{Washer} \]
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\[ V=s\pi\int_a^brhdx\tag{Shell} \]
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\[ V=\int_a^bAdx\tag{Cross Section} \]
Series
\[
error\leq|a_{n+1}|\tag{Alt. Series Error}
\]
\[
error\leq\left|\frac{f^{(n+1)}(z)(x-c)^{n+1}}{(n+1)!}\right|\tag{Lagrange Error}
\]
\[
e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots+\frac{x^n}{n!}+\ldots
\]
\[
\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots+\frac{(-1)^nx^{2n+1}}{(2n+1)!}+\ldots
\]
\[
\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots+\frac{(-1)^nx^{2n}}{(2n)!}+\ldots
\]
\[
\ln(x)=(x-1)-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}-\dots+\frac{(-1)^{n+1}(x-1)^n}{n}+\ldots
\]
\[
f(x)=f(c)+f'(c)(x-c)+\frac{f''(c)}{2!}(x-c)^2+\ldots+\frac{f^{(n)}(c)}{n!}(x-c)^n+\ldots\tag{Taylor Series}
\]
\[
f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\ldots+\frac{f^{(n)}(0)}{n!}x^n+\ldots\tag{Maclaurin Series}
\]
Tests for Convergence/Divergence
\[
diverges\hspace{4pt}if \lim_{n\to\infty}a_n\neq0\tag{$n^{th}$ term test}
\]
\[
\sum_{n=0}^\infty ar^n\hspace{12pt}|r|<1:conv., |r|\geq1:div., S=\frac{a}{1-r}\tag{Geometric series test} \]
\[ \sum_{n=1}^\infty \frac{1}{n^p}\hspace{12pt}p>1:conv.,p\leq1:div.\tag{p-series}
\]
\[
decr.\hspace{4pt}terms\hspace{4pt}and \lim_{n\to\infty}a_n=0\to conv.\tag{Alternating series}
\]
\[
a_n=f(x)\hspace{12pt}\sum_{n=1}^{\infty}a_n \hspace{4pt} conv.\hspace{4pt}if\int_1^\infty f(x)dx \hspace{4pt} conv.,\sum_{n=1}^\infty a_n\hspace{4pt}div. \hspace{4pt}if \int_1^\infty f(x)dx\hspace{4pt}div.\tag{Integral Test}
\]
\[
\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|<1\to conv., \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|>1\to div., \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=1\to inconclusive\tag{Ratio Test}
\]
\[
0\leq a_n\leq b_n\hspace{12pt} \sum_{n=1}^\infty b_n\hspace{4pt}conv.\to \sum_{n=1}^\infty a_n\hspace{4pt}conv.,\hspace{12pt}\sum_{n=1}^\infty a_n\hspace{4pt}div.\to \sum_{n=1}^\infty b_n\hspace{4pt}div.\tag{Direct Comparison}
\]
\[
\lim_{n\to\infty}\frac{a_n}{b_n} \hspace{4pt} finite\hspace{4pt}and\hspace{4pt}positive\to both\hspace{4pt}conv.\hspace{4pt}or\hspace{4pt}div.\tag{Limit Comparison}
\]
Parametric Equations and Polar Coordinates
\[
S=\int_a^b\sqrt{1+(f'(x))^2}dx=\int_{t_1}^{t_2}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt=\int_{\theta_1}^{\theta_2}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta\tag{Arc Length}
\]
\[
|\vec{v}(t)|=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\tag{Speed}
\]
\[
\int_{t_1}^{t_2}|\vec{v}(t)|dt=\int_{t_1}^{t_2}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt\tag{Total Distance}
\]
\[
\frac{1}{2}\int_{\theta_1}^{\theta_2}r^2d\theta\tag{Polar Area}
\]