INTEGRALS CONTAINING Tan(ax)

1.

$\displaystyle\int\tan ax dx=-\displaystyle \frac{1}{a}\ln\cos ax=\displaystyle \frac{1}{a}\ln\sec ax$

2.

$\displaystyle\int\tan^2 ax dx=\displaystyle \frac{\tan ax}{a}-x$

3.

$\displaystyle\int\tan^3 ax dx=\displaystyle \frac{\tan^2 ax}{2a}+\displaystyle \frac{1}{a}\ln\cos ax$

4.

$\displaystyle\int\tan^n ax \sec^2 ax dx=\displaystyle \frac{\tan^{n+1}ax}{(n+1)a}$

5.

$\displaystyle\int\displaystyle \frac{\sec^2 ax}{\tan ax}dx=\displaystyle \frac{1}{a}\ln\tan ax$

6.

$\displaystyle\int\displaystyle \frac{dx}{\tan ax}=\displaystyle \frac{1}{a}\ln\sin ax$

7.

$\displaystyle\int x\tan ax dx=\displaystyle \frac{1}{a^2}\left\{\displaystyle \... ...ystyle \frac{2^{2n}(2^{2n}-1)B_n(ax)^{2n+1}}{(2n+1)!}+ \cdot\cdot\cdot \right\}$

where the constants B_n are the Bernoulli's numbers.

8.

$\displaystyle\int\displaystyle \frac{\tan ax}{x}dx=ax+\displaystyle \frac{(ax)^... ...isplaystyle \frac{2^{2n}(2^{2n}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdot\cdot\cdot$

where the constants B_n are the Bernoulli's numbers.

9.

$\displaystyle\int x\tan^2 ax dx=\displaystyle \frac{x\tan ax}{a}+\displaystyle \frac{1}{a^2}\ln\cos ax-\displaystyle \frac{x^2}{2}$

10.

$\displaystyle\int\displaystyle \frac{dx}{p+q\tan ax}=\displaystyle \frac{px}{p^2+q^2}+\displaystyle \frac{q}{a(p^2+q^2)}\ln(q\sin ax+p\cos ax)$

11.

$\displaystyle\int\tan^n ax dx=\displaystyle \frac{\tan^{n-1}ax}{(n-1)a}-\int\tan^{n-2}ax dx$