\end{tabularx}
\vfill
+\vtop to 5cm {
+ \flushbottom
+ \subsubsection*{Index identities}
+ \begin{align*}
+ a^{x+y} &= a^x \cdot a^y \\
+ a^{x-y} &= a^x \div a^y \\
+ (a^x)^y &= a^{x \cdot y} \\
+ (a \cdot b)^x &= a^x \cdot b^x
+ \end{align*}
+}
+
\subsection*{Antiderivatives}
\(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
\hline
\end{tabularx}
+\rowcolors{2}{white}{white}
\vspace{1em}
Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)\) is constant \(\forall \> x \in (-a, a)\)
+
+\vfill
+\vtop to 5cm {
+ \flushbottom
+ \subsubsection*{Logarithmic identities}
+ \begin{align*}
+ \log_b (xy) &= \log_b x + \log_b y \\
+ \log_b\left(\frac{x}{y}\right) &= \log_b(x) - \log_b(y) \\
+ \log_b y^{x^n} &= x^n \log_b y \\
+ \log_b x^n &= n \log_b x
+ \end{align*}
+}