1.$$\int \frac{dx}{x^2+a^2} = \frac{1}{a} \arctan{\frac{x}{a}}+C$$

2.$$\int \frac{dx}{x^2-a^2}=\frac{1}{2a} \ln{\lvert \frac{x-a}{x+a} \rvert}+C$$

3.$$\int \frac{dx}{\sqrt{a^2-x^2}}=\arcsin{\frac{x}{a}}+C$$

4.$$\int \frac{dx}{\sqrt{x^2+a^2}}=\ln{\lvert x+\sqrt{x^2+a^2} \rvert}+C$$

5.$$\int \ln{x} dx =x\ln{x} -x+C$$

6.$$\int e^{ax} \cos{bx}dx = \frac{e^{ax}}{a^2+b^2}(a\cos{bx}+b\sin{bx})+C$$

7.$$\int e^{ax} \sin{bx}dx = \frac{e^{ax}}{a^2+b^2}(a\sin{bx}-b\cos{bx})+C$$

8.$$\int x\cos{nx}dx = \frac{1}{n^2}\cos{nx}+\frac{1}{n}\sin{nx}+C$$

9.$$\int x\sin{nx}dx = \frac{1}{n^2}\sin{nx}-\frac{1}{n}\cos{nx}+C$$

10.$$\int \sec{x}dx = ln{\lvert \sec{x}+\tan{x} \rvert}+C$$

11.$$
\begin{equation}
\begin{split}
I_n &=\int \tan^n{x} dx \quad [\tan^n{x} = \tan^{n-2}{x}(\sec^{2}{x}-1)] \\\\
&= \frac{1}{n} \tan^{n-1}{x}\ -\ I_{n-2}
\end{split}
\nonumber
\end{equation}
$$

12.$$
\begin{equation}
\begin{split}
I_n &=\int \ln^{n}{x} dx \\\\
&= x \ln^{n}{x}\ -\ nI_{n-1}
\end{split}
\nonumber
\end{equation}
$$