(1) \( 1/x \) equations
$$\int \frac{1}{x} \, dx = \ln |x|$$
$$\int \frac{1}{ax+b} \, dx = \frac{1}{a} \ln |ax+b|$$
(2) \( e \) equations
$$\int e^x \, dx = e^x$$
$$\int e^{ax+b} \, dx = \frac{1}{a} e^{ax+b}$$
(3) Integrating to atan(x)
$$\int \frac{1}{ax^2+b} \, dx$$
Example:
$$\int \frac{1}{3x^2+2} \, dx$$
Step 1: Convert \( b \) to 1
$$\int \frac{1}{2} \left( \frac{1}{\frac{3}{2} x^2+1} \right) \, dx = \frac{1}{2} \int \frac{1}{\frac{3}{2} x^2+1} \, dx$$
Step 2: Square root \( (ax^2) \)
$$\frac{1}{2} \int \frac{1}{\left(\sqrt{\frac{3}{2}} x\right)^2+1} \, dx = \frac{1}{2} \int \frac{1}{\left(\sqrt{\frac{3}{2}} x\right)^2+1} \, dx$$
Step 3: Integrate
$$\frac{1}{\sqrt{\frac{3}{2}}} \cdot \frac{1}{\sqrt{\frac{3}{2}}} \tan^{-1}\left(\sqrt{\frac{3}{2}} x\right) = \frac{\sqrt{2}}{2\sqrt{3}} \tan^{-1}\left(\sqrt{\frac{3}{2}} x\right)$$
(4) Trigonometry
$$\begin{array}{|c|c|}\hline \sin x, \sin(ax+b) & - \cos x, -\frac{1}{a}\cos(ax+b)\\ \hline \cos x, \cos(ax+b) & \sin x, \frac{1}{a}\sin(ax+b)\\ \hline \sec x, \sec(ax+b) & \tan x, \frac{1}{a}\tan(ax+b) \\ \hline \end{array}$$