Quick reference: common integrals and integration techniques

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\title{A Quick Reference for Common Integrals}
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\maketitle

\section*{Power and elementary functions}
\begin{align*}
  \int x^n \, dx &= \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \\
  \int \frac{1}{x} \, dx &= \ln |x| + C \\
  \int e^x \, dx &= e^x + C \\
  \int a^x \, dx &= \frac{a^x}{\ln a} + C \quad (a > 0,\, a \neq 1)
\end{align*}

\section*{Trigonometric functions}
\begin{align*}
  \int \sin x \, dx &= -\cos x + C \\
  \int \cos x \, dx &= \sin x + C \\
  \int \tan x \, dx &= -\ln |\cos x| + C \\
  \int \sec^2 x \, dx &= \tan x + C \\
  \int \frac{1}{\sqrt{1 - x^2}} \, dx &= \arcsin x + C \\
  \int \frac{1}{1 + x^2} \, dx &= \arctan x + C
\end{align*}

\section*{Integration techniques}

\subsection*{1. Integration by parts}
\[
  \int u \, dv = uv - \int v \, du
\]
Choice mnemonic (LIATE): \emph{Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential} --- pick $u$ from the higher class.

\subsection*{2. Substitution}
If $u = g(x)$, then $du = g'(x)\, dx$ and
\[
  \int f(g(x))\, g'(x)\, dx = \int f(u)\, du.
\]

\subsection*{3. Partial fractions}
For a rational function $\dfrac{P(x)}{Q(x)}$ with $\deg P < \deg Q$, decompose $Q$ into linear and quadratic factors and write the integrand as a sum of simpler fractions.

\section*{Worked example}
Compute $\displaystyle \int x e^{-x}\, dx$.

By parts with $u = x$, $dv = e^{-x} dx$, so $du = dx$ and $v = -e^{-x}$:
\[
  \int x e^{-x} \, dx
  = -x e^{-x} - \int (-e^{-x}) \, dx
  = -x e^{-x} - e^{-x} + C
  = -(x+1)\, e^{-x} + C.
\]

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