INTEGRALS CONTAINING Csc(ax)

1.

$\displaystyle\int\csc axdx=\displaystyle \frac{1}{a}\ln(\csc ax-\cot ax)=\displaystyle \frac{1}{a}\ln\tan\displaystyle \frac{ax}{2}$

2.

$\displaystyle\int\csc^2 axdx=-\displaystyle \frac{\cot ax}{a}$

3.

$\displaystyle\int\csc^3 axdx=-\displaystyle \frac{\csc ax\cot ax}{2a}+\displaystyle \frac{1}{2a}\ln\tan\displaystyle \frac{ax}{2}$

4.

$\displaystyle\int\csc^n ax\cot axdx=-\displaystyle \frac{\csc^n ax}{na}$

5.

$\displaystyle\int\displaystyle \frac{dx}{\csc ax}=-\displaystyle \frac{\cos ax}{a}$

6.

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

where the constants B_n are the Bernoulli's numbers.

7.

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

where the constants B_n are the Bernoulli's numbers.

8.

$\displaystyle\int x\csc^2 axdx=-\displaystyle \frac{x\cot ax}{a}+\displaystyle \frac{1}{a^2}\ln\sin ax$

9.

$\displaystyle\int\displaystyle \frac{dx}{q+p\csc ax}=\displaystyle \frac{x}{q}-\displaystyle \frac{p}{q}\int\displaystyle \frac{dx}{p+q\sin ax}$

10.

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