INTEGRALS CONTAINING Sec(ax)

1.

$\displaystyle\int\sec ax dx=\displaystyle \frac{1}{a}\ln(\sec ax+\tan ax)=\disp... ...1}{a}\ln\tan\left(\displaystyle \frac{ax}{2}+\displaystyle \frac{\pi}{4}\right)$

2.

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

3.

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

4.

$\displaystyle\int \sec^n ax \tan axdx=\displaystyle \frac{\sec^n ax}{na}$

5.

$\displaystyle\int \displaystyle \frac{dx}{\sec ax}=\displaystyle \frac{sin ax}{a}$

6.

$\displaystyle\int x\sec ax dx=\displaystyle \frac{1}{a^2}\left( \displaystyle \... ...t + \displaystyle \frac{E_{n}(ax)^{2n+2}}{(2n+2)(2n)!}+ \cdot\cdot\cdot \right)$

where the constants E_n are the Euler's numbers.

7.

$\displaystyle\int\displaystyle \frac{\sec ax}{x}dx=\ln x+\displaystyle \frac{(a... ...\cdot\cdot\cdot + \displaystyle \frac{E_{n}(ax)^{2n}}{2n(2n)!}+ \cdot\cdot\cdot$

where the constants E_n are the Euler's numbers.

8.

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

9.

$\displaystyle\int\displaystyle \frac{dx}{q+p\sec ax}=\displaystyle \frac{x}{q}-\displaystyle \frac{p}{q}\int\displaystyle \frac{dx}{p+q\cos ax}$

10.

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