\section{Integral Tables}
\label{app:integral_tables}

This appendix provides commonly used integral formulas encountered throughout the handbook. See \cref{ch:first_order} for separable equations, \cref{ch:second_order_homogeneous} and \cref{ch:second_order_nonhomogeneous} for second-order ODE techniques, \cref{ch:laplace_transforms} for Laplace-domain integration, and \cref{ch:fourier_series} for trigonometric integrals in series expansions.

\subsection{Basic Integrals}
\label{sec:appC_basic}

\begin{table}[htbp]
\centering
\caption{Basic Integral Formulas}
\label{tab:integrals_basic}
\begin{tabular}{l l}
\toprule
\textbf{Integrand} & \textbf{Result} \\
\midrule
$\displaystyle \int x^n \, dx$ & $\displaystyle \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ \\[8pt]
$\displaystyle \int \frac{1}{x} \, dx$ & $\displaystyle \ln|x| + C$ \\[8pt]
$\displaystyle \int e^{ax} \, dx$ & $\displaystyle \frac{e^{ax}}{a} + C$ \\[8pt]
$\displaystyle \int a^x \, dx$ & $\displaystyle \frac{a^x}{\ln(a)} + C$ \\[8pt]
$\displaystyle \int \sin(ax) \, dx$ & $\displaystyle -\frac{\cos(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \cos(ax) \, dx$ & $\displaystyle \frac{\sin(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \tan(ax) \, dx$ & $\displaystyle -\frac{\ln|\cos(ax)|}{a} + C = \frac{\ln|\sec(ax)|}{a} + C$ \\[8pt]
$\displaystyle \int \sec^2(ax) \, dx$ & $\displaystyle \frac{\tan(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \csc^2(ax) \, dx$ & $\displaystyle -\frac{\cot(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \sec(ax)\tan(ax) \, dx$ & $\displaystyle \frac{\sec(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \csc(ax)\cot(ax) \, dx$ & $\displaystyle -\frac{\csc(ax)}{a} + C$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\subsection{Integrals Involving Exponential and Trigonometric Functions}
\label{sec:appC_exptig}

\begin{table}[htbp]
\centering
\caption{Exponential--Trigonometric Integrals}
\label{tab:integrals_exptig}
\begin{tabular}{l l}
\toprule
\textbf{Integrand} & \textbf{Result} \\
\midrule
$\displaystyle \int e^{ax}\sin(bx)\,dx$ & $\displaystyle \frac{e^{ax}\bigl(a\sin(bx) - b\cos(bx)\bigr)}{a^2 + b^2} + C$ \\[8pt]
$\displaystyle \int e^{ax}\cos(bx)\,dx$ & $\displaystyle \frac{e^{ax}\bigl(a\cos(bx) + b\sin(bx)\bigr)}{a^2 + b^2} + C$ \\[8pt]
$\displaystyle \int x e^{ax}\,dx$ & $\displaystyle e^{ax}\!\left(\frac{x}{a} - \frac{1}{a^2}\right) + C$ \\[8pt]
$\displaystyle \int x \sin(ax)\,dx$ & $\displaystyle \frac{\sin(ax)}{a^2} - \frac{x\cos(ax)}{a} + C$ \\[8pt]
$\displaystyle \int x \cos(ax)\,dx$ & $\displaystyle \frac{\cos(ax)}{a^2} + \frac{x\sin(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \ln(x)\,dx$ & $\displaystyle x\ln(x) - x + C$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\subsection{Integrals Involving Inverse Trigonometric Functions}
\label{sec:appC_inverse_trig}

\begin{table}[htbp]
\centering
\caption{Inverse Trigonometric Integrals}
\label{tab:integrals_inverse_trig}
\begin{tabular}{l l}
\toprule
\textbf{Integrand} & \textbf{Result} \\
\midrule
$\displaystyle \int \frac{1}{\sqrt{a^2 - x^2}}\,dx$ & $\displaystyle \arcsin\!\left(\frac{x}{a}\right) + C$ \\[8pt]
$\displaystyle \int \frac{1}{a^2 + x^2}\,dx$ & $\displaystyle \frac{1}{a}\arctan\!\left(\frac{x}{a}\right) + C$ \\[8pt]
$\displaystyle \int \frac{1}{x\sqrt{x^2 - a^2}}\,dx$ & $\displaystyle \frac{1}{a}\,\text{arcsec}\!\left(\frac{|x|}{a}\right) + C$ \\[8pt]
$\displaystyle \int \sqrt{a^2 - x^2}\,dx$ & $\displaystyle \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\arcsin\!\left(\frac{x}{a}\right) + C$ \\[8pt]
$\displaystyle \int \frac{1}{\sqrt{x^2 + a^2}}\,dx$ & $\displaystyle \ln\!\bigl|x + \sqrt{x^2 + a^2}\,\bigr| + C = \text{arcsinh}\!\left(\frac{x}{a}\right) + C$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\subsection{Hyperbolic Integrals}
\label{sec:appC_hyperbolic}

\begin{table}[htbp]
\centering
\caption{Hyperbolic Integral Formulas}
\label{tab:integrals_hyperbolic}
\begin{tabular}{l l}
\toprule
\textbf{Integrand} & \textbf{Result} \\
\midrule
$\displaystyle \int \sinh(ax)\,dx$ & $\displaystyle \frac{\cosh(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \cosh(ax)\,dx$ & $\displaystyle \frac{\sinh(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \tanh(ax)\,dx$ & $\displaystyle \frac{\ln|\cosh(ax)|}{a} + C$ \\[8pt]
$\displaystyle \int \text{sech}^2(ax)\,dx$ & $\displaystyle \frac{\tanh(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \text{csch}^2(ax)\,dx$ & $\displaystyle -\frac{\coth(ax)}{a} + C$ \\[8pt]
$\displaystyle \int \text{sech}(ax)\tanh(ax)\,dx$ & $\displaystyle -\frac{\text{sech}(ax)}{a} + C$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\subsection{Additional Common Integrals}
\label{sec:appC_additional}

\begin{table}[htbp]
\centering
\caption{Additional Common Integral Formulas}
\label{tab:integrals_additional}
\begin{tabular}{l l}
\toprule
\textbf{Integrand} & \textbf{Result} \\
\midrule
$\displaystyle \int \frac{1}{x^2 - a^2}\,dx$ & $\displaystyle \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$ \\[8pt]
$\displaystyle \int \frac{1}{a^2 - x^2}\,dx$ & $\displaystyle \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C$ \\[8pt]
$\displaystyle \int \frac{x}{\sqrt{a^2 + x^2}}\,dx$ & $\displaystyle \sqrt{a^2 + x^2} + C$ \\[8pt]
$\displaystyle \int \frac{dx}{x\sqrt{a^2 + x^2}}$ & $\displaystyle -\frac{1}{a}\ln\left|\frac{a + \sqrt{a^2 + x^2}}{|x|}\right| + C$ \\[8pt]
$\displaystyle \int \sqrt{x^2 + a^2}\,dx$ & $\displaystyle \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\ln\!\bigl|x + \sqrt{x^2 + a^2}\,\bigr| + C$ \\[8pt]
$\displaystyle \int \sqrt{x^2 - a^2}\,dx$ & $\displaystyle \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\ln\!\bigl|x + \sqrt{x^2 - a^2}\,\bigr| + C$ \\[8pt]
\bottomrule
\end{tabular}
\end{table}

\subsection{Integration by Parts Formula}
\label{sec:appC_parts}

The method of integration by parts, frequently used in solving ODEs (\cref{ch:first_order}), is based on the product rule:
\begin{equation}
\label{eq:integration_by_parts}
\int u \, dv = uv - \int v \, du.
\end{equation}
Choose $u$ and $dv$ so that $\int v \, du$ is simpler than the original integral.

\subsection{Reduction Formulas}
\label{sec:appC_reduction}

Reduction formulas allow integrals of higher powers to be expressed in terms of lower powers:

\begin{align}
\label{eq:reduction_sin}
\int \sin^n(x) \, dx &= -\frac{1}{n}\sin^{n-1}(x)\cos(x) + \frac{n-1}{n}\int \sin^{n-2}(x) \, dx, \\
\label{eq:reduction_cos}
\int \cos^n(x) \, dx &= \frac{1}{n}\cos^{n-1}(x)\sin(x) + \frac{n-1}{n}\int \cos^{n-2}(x) \, dx, \\
\label{eq:reduction_xe}
\int x^n e^{ax} \, dx &= \frac{x^n e^{ax}}{a} - \frac{n}{a}\int x^{n-1}e^{ax} \, dx.
\end{align}

These formulas are especially useful when solving higher-order ODEs (\cref{ch:second_order_homogeneous,ch:second_order_nonhomogeneous}) and evaluating Fourier coefficients (\cref{ch:fourier_series}).
