INTEGRALS CONTAINING Cot(ax)

1.

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

2.

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

3.

$\displaystyle\int \cot^3 ax dx=-\displaystyle \frac{\cot^2 ax}{2a}-\displaystyle \frac{1}{a}\ln\sin ax$

4.

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

5.

$\displaystyle\int\displaystyle \frac{\csc^2ax}{\cot ax}dx=-\displaystyle \frac{1}{a}\ln\cot ax$

6.

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

7.

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

where the constants B_n are the Bernoulli's numbers.

8.

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

where the constants B_n are the Bernoulli's numbers.

9.

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

10.

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

11.

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

12.

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