INTEGRALS CONTAINING Cot(ax)

1.

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

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

where the constants B_n are the Bernoulli's numbers.

8.

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

where the constants B_n are the Bernoulli's numbers.

9.

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

10.

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

11.

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

12.

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

[Tables]

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