diff --git a/appendices/appC_integral_tables.tex b/appendices/appC_integral_tables.tex
index 4a582cb..46ead80 100644
--- a/appendices/appC_integral_tables.tex
+++ b/appendices/appC_integral_tables.tex
@@ -1,7 +1,7 @@
 \section{Integral Tables}
 \label{app:integral_tables}
 
-This appendix provides commonly used integral formulas encountered throughout the handbook.
+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}
@@ -14,17 +14,17 @@ This appendix provides commonly used integral formulas encountered throughout th
 \toprule
 \textbf{Integrand} & \textbf{Result} \\
 \midrule
-$\displaystyle \int x^n \, dx$ & TBD \\
-$\displaystyle \int \frac{1}{x} \, dx$ & TBD \\
-$\displaystyle \int e^{ax} \, dx$ & TBD \\
-$\displaystyle \int a^x \, dx$ & TBD \\
-$\displaystyle \int \sin(ax) \, dx$ & TBD \\
-$\displaystyle \int \cos(ax) \, dx$ & TBD \\
-$\displaystyle \int \tan(ax) \, dx$ & TBD \\
-$\displaystyle \int \sec^2(ax) \, dx$ & TBD \\
-$\displaystyle \int \csc^2(ax) \, dx$ & TBD \\
-$\displaystyle \int \sec(ax)\tan(ax) \, dx$ & TBD \\
-$\displaystyle \int \csc(ax)\cot(ax) \, dx$ & TBD \\
+$\displaystyle \int x^n \, dx$ & $\displaystyle \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ \\[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}
@@ -40,12 +40,12 @@ $\displaystyle \int \csc(ax)\cot(ax) \, dx$ & TBD \\
 \toprule
 \textbf{Integrand} & \textbf{Result} \\
 \midrule
-$\displaystyle \int e^{ax}\sin(bx)\,dx$ & TBD \\
-$\displaystyle \int e^{ax}\cos(bx)\,dx$ & TBD \\
-$\displaystyle \int x e^{ax}\,dx$ & TBD \\
-$\displaystyle \int x \sin(ax)\,dx$ & TBD \\
-$\displaystyle \int x \cos(ax)\,dx$ & TBD \\
-$\displaystyle \int \ln(x)\,dx$ & TBD \\
+$\displaystyle \int e^{ax}\sin(bx)\,dx$ & $\displaystyle \frac{e^{ax}\bigl(a\sin(bx) - b\cos(bx)\bigr)}{a^2 + b^2} + C$ \\[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}
@@ -61,11 +61,79 @@ $\displaystyle \int \ln(x)\,dx$ & TBD \\
 \toprule
 \textbf{Integrand} & \textbf{Result} \\
 \midrule
-$\displaystyle \int \frac{1}{\sqrt{a^2 - x^2}}\,dx$ & TBD \\
-$\displaystyle \int \frac{1}{a^2 + x^2}\,dx$ & TBD \\
-$\displaystyle \int \frac{1}{x\sqrt{x^2 - a^2}}\,dx$ & TBD \\
-$\displaystyle \int \sqrt{a^2 - x^2}\,dx$ & TBD \\
-$\displaystyle \int \frac{1}{\sqrt{x^2 + a^2}}\,dx$ & TBD \\
+$\displaystyle \int \frac{1}{\sqrt{a^2 - x^2}}\,dx$ & $\displaystyle \arcsin\!\left(\frac{x}{a}\right) + C$ \\[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}).