INTEGRALES CONTAINING Sin(ax)

1.

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

2.

$\displaystyle \int x\sin ax\,dx=\displaystyle \frac{\sin ax}{a^{\displaystyle2}}\,-\,\displaystyle \frac{x\cos ax}{a}$

3.

$\displaystyle \int x^{\displaystyle2}\sin ax\,dx=\displaystyle \frac{2x}{a^{\di... ...^{\displaystyle3}}\,-\,\displaystyle \frac{x^{\displaystyle2}}{a}\right)\cos ax$

4.

$\displaystyle \int\displaystyle \frac{\sin ax}{x}\,dx=ax\,-\,\displaystyle \fra... ...3\cdot 3!}\,+\,\displaystyle \frac{(ax)^{\displaystyle5}}{5\cdot 5!}\,-\,\cdots$

5.

$\displaystyle \int\displaystyle \frac{\sin ax}{x^{\displaystyle2}}\,dx=-\displa... ...le \frac{\sin ax}{x}\,+\,a\displaystyle \int\displaystyle \frac{\cos ax}{x}\,dx$

6.

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

7.

$\displaystyle \int\displaystyle \frac{x\,dx}{\sin ax}=\displaystyle \frac{1}{a^... ...\displaystyle2n-1}-1)B_{n}(ax)^{\displaystyle2n+1}}{(2n+1)!}\,+\,\cdots\right\}$

where the constants B_n are the Bernoulli's numbers.

8.

$\displaystyle \int\sin^{\displaystyle2}ax\,dx=\displaystyle \frac{x}{2}\,-\,\displaystyle \frac{\sin 2ax}{4a}$

9.

$\displaystyle \int x\sin^{\displaystyle2} ax\,dx=\displaystyle \frac{x^{\displa... ...yle \frac{x\sin 2ax}{4a}\,-\,\displaystyle \frac{\cos 2ax}{8a^{\displaystyle2}}$

10.

$\displaystyle \int\sin^{\displaystyle3} ax\,dx=-\displaystyle \frac{\cos ax}{a}\,+\,\displaystyle \frac{cos^{\displaystyle3}ax}{3a}$

11.

$\displaystyle \int\sin^{\displaystyle4} ax\,dx=\displaystyle \frac{3x}{8}\,-\,\displaystyle \frac{\sin 2ax}{4a}\,+\,\displaystyle \frac{\sin 4ax}{32a}$

12.

$\displaystyle \int\displaystyle \frac{dx}{\sin ^{\displaystyle2}ax}=-\displaystyle \frac{1}{a}\cot ax$

13.

$\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle3} ax}=-\displayst... ...splaystyle2}ax}\,+\,\displaystyle \frac{1}{2a}\ln\tan\displaystyle \frac{ax}{2}$

14.

$\displaystyle \int\sin px\sin qx\,dx=\displaystyle \frac{\sin(p-q)x}{2(p-q)}\,-\,\displaystyle \frac{\sin(p+q)x}{2(p+q)}$
provided $p \neq \pm q$ .

15.

$\displaystyle \int\displaystyle \frac{dx}{1-\sin ax}=\displaystyle \frac{1}{a}\tan\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)$

16.

$\displaystyle \int\displaystyle \frac{x\,dx}{1-\sin ax}=\displaystyle \frac{x}{... ...}\ln\sin\left(\displaystyle \frac{\pi}{4}\,-\,\displaystyle \frac{ax}{2}\right)$

17.

$\displaystyle \int\displaystyle \frac{dx}{1+\sin ax}=-\displaystyle \frac{1}{a}\tan\left(\displaystyle \frac{\pi}{4}\,-\,\displaystyle \frac{ax}{2}\right)$

18.

$\displaystyle \int\displaystyle \frac{x\,dx}{1+\sin ax}=-\displaystyle \frac{x}... ...}\ln\sin\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)$

19.

$\displaystyle \int\displaystyle \frac{dx}{(1-\sin ax)^{\displaystyle2}}=\displa... ...ystyle3}\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)$

20.

$\displaystyle \int\displaystyle \frac{dx}{(1+\sin ax)^{\displaystyle2}}=-\displ... ...ystyle3}\left(\displaystyle \frac{\pi}{4}\,-\,\displaystyle \frac{ax}{2}\right)$

21.

$\displaystyle \int\displaystyle \frac{dx}{p+q\sin ax}=\left\{\begin{array}{l} ... ...style \sqrt{q^{\displaystyle2}-p^{\displaystyle2}}}\right) \end{array} \right.$

22.

$\displaystyle \int\displaystyle \frac{dx}{(p+q\sin ax)^{\displaystyle2}}=\displ... ...tyle2}-q^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{p+q\sin ax}$

23.

$\displaystyle \int\displaystyle \frac{dx}{p^{\displaystyle2}+q^{\displaystyle2}... ...yle \frac{\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}}\tan ax}{p}$

24.

$\displaystyle \int\displaystyle \frac{dx}{p^{\displaystyle2}-q^{\displaystyle2}... ...rt{q^{\displaystyle2}-p^{\displaystyle2}}\tan ax-p}\right) \end{array} \right.$

25.

$\displaystyle \int x^{\displaystyle m}\sin ax\,dx=-\displaystyle \frac{x^{\disp... ...m(m-1)}{a^{\displaystyle2}}\displaystyle \int\ x^{\displaystyle m-2}\sin ax\,dx$

26.

$\displaystyle \int\displaystyle \frac{\sin ax}{x^{\displaystyle n}}\,dx=-\displ... ...}{n-1}\displaystyle \int\displaystyle \frac{\cos ax}{x^{\displaystyle n-1}}\,dx$

27.

$\displaystyle \int\sin^{\displaystyle n}ax\,dx=-\displaystyle \frac{\sin^{\disp... ...,+\,\displaystyle \frac{n-1}{n}\displaystyle \int\sin^{\displaystyle n-2}ax\,dx$

28.

$\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle n}ax}=\displaysty... ...{n-2}{n-1}\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle n-2}ax}$

29.

$\displaystyle \int\displaystyle \frac{x\,dx}{\sin^{\displaystyle n}ax}=\display... ...2}{n-1}\displaystyle \int\displaystyle \frac{x\,dx}{\sin^{\displaystyle n-2}ax}$