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

[Tables]

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