INTEGRALS CONTAINING Sin(ax) And Cos(ax)

1.

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

2.

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

3.

$\displaystyle \int\sin^{\displaystyle n}ax\cos ax\,dx=\displaystyle \frac{\sin^{\displaystyle n+1}ax}{(n+1)a}$

4.

$\displaystyle \int\cos^{\displaystyle n}ax\sin ax\,dx=-\displaystyle \frac{\cos^{\displaystyle n+1}ax}{(n+1)a}$

5.

$\displaystyle \int\sin^{\displaystyle2}ax\cos^{\displaystyle2}ax\,dx=\displaystyle \frac{x}{8}\,-\,\displaystyle \frac{\sin 4ax}{32a}$

6.

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

7.

$\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle2}ax\cos ax}=\disp... ...}{4}\,+\,\displaystyle \frac{ax}{2}\right)\,-\,\displaystyle \frac{1}{a\sin ax}$

8.

$\displaystyle \int\displaystyle \frac{dx}{\sin ax\cos^{\displaystyle2}ax}=\disp... ...rac{1}{a}\ln\tan\displaystyle \frac{ax}{2}\,+\,\displaystyle \frac{1}{a\cos ax}$

9.

$\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle2}ax\cos^{\displaystyle2}ax}=-\displaystyle \frac{2\cot 2ax}{a}$

10.

$\displaystyle \int\displaystyle \frac{\sin^{\displaystyle2}ax}{\cos ax}\,dx=-\d... ...}\ln\tan\left(\displaystyle \frac{ax}{2}\,+\,\displaystyle \frac{\pi}{4}\right)$

11.

$\displaystyle \int\displaystyle \frac{\cos^{\displaystyle2}ax}{\sin ax}\,dx=\di... ...frac{\cos ax}{a}\,+\,\displaystyle \frac{1}{a}\ln\tan\displaystyle \frac{ax}{2}$

12.

$\displaystyle \int\displaystyle \frac{dx}{\cos ax(1\pm\sin ax)}=\mp\displaystyl... ...}\ln\tan\left(\displaystyle \frac{ax}{2}\,+\,\displaystyle \frac{\pi}{4}\right)$

13.

$\displaystyle \int\displaystyle \frac{dx}{\sin ax(1\pm\cos ax)}=\pm\displaystyl... ...a(1\pm\cos ax)}\,+\,\displaystyle \frac{1}{2a}\ln\tan\displaystyle \frac{ax}{2}$

14.

$\displaystyle \int\displaystyle \frac{dx}{\sin ax\pm\cos ax}=\displaystyle \fra... ...ln\tan\left(\displaystyle \frac{ax}{2}\,\pm\,\displaystyle \frac{\pi}{8}\right)$

15.

$\displaystyle \int\displaystyle \frac{\sin ax\,dx}{\sin ax\pm\cos ax}=\displaystyle \frac{x}{2}\mp\displaystyle \frac{1}{2a}\ln(\sin ax\pm\cos ax)$

16.

$\displaystyle \int\displaystyle \frac{\cos ax\,dx}{\sin ax\pm\cos ax}=\pm\displaystyle \frac{x}{2}\,+\,\displaystyle \frac{1}{2a}\ln(\sin ax\pm\cos ax)$

17.

$\displaystyle \int\displaystyle \frac{\sin ax\,dx}{p+q\cos ax}=-\displaystyle \frac{1}{aq}\ln(p+q\cos ax)$

18.

$\displaystyle \int\displaystyle \frac{\cos ax\,dx}{p+q\sin ax}=\displaystyle \frac{1}{aq}\ln(p+q\sin ax)$

19.

$\displaystyle \int\displaystyle \frac{\sin ax\,dx}{(p+q\cos ax)^{\displaystyle n}}=\displaystyle \frac{1}{aq(n-1)(p+q\cos ax)^{\displaystyle n-1}}$

20.

$\displaystyle \int\displaystyle \frac{\cos ax\,dx}{(p+q\sin ax)^{\displaystyle n}}=\displaystyle \frac{-1}{aq(n-1)(p+q\sin ax)^{\displaystyle n-1}}$

21.

$\displaystyle \int\displaystyle \frac{dx}{p\sin ax+q\cos ax}=\displaystyle \fra... ...e2}}}\ln\tan\left(\displaystyle \frac{ax+\tan^{\displaystyle-1}(q/p)}{2}\right)$

22.

$\displaystyle \int\displaystyle \frac{dx}{p\sin ax+q\cos ax+r}=\left\{\begin{ar... ...displaystyle2}-r^{\displaystyle2}}+(r-q)\tan(ax/2)}\right) \end{array} \right.$

23.

$\displaystyle \int\displaystyle \frac{dx}{p\sin ax+q(1+\cos ax)}=\displaystyle \frac{1}{ap}\ln\left(q+p\tan\displaystyle \frac{ax}{2}\right)$

24.

$\displaystyle \int\displaystyle \frac{dx}{p\sin ax+q\cos ax\pm\displaystyle \sq... ...rac{\pi}{4}\,\mp\,\displaystyle \frac{ax+\tan^{\displaystyle-1}(q/p)}{2}\right)$

25.

$\displaystyle \int\displaystyle \frac{dx}{p^{\displaystyle2}\sin^{\displaystyle... ...frac{1}{apq}\tan^{\displaystyle-1}\left(\displaystyle \frac{p\tan ax}{q}\right)$

26.

$\displaystyle \int\displaystyle \frac{dx}{p^{\displaystyle2}\sin^{\displaystyle... ...style \frac{1}{2apq}\ln\left(\displaystyle \frac{p\tan ax-q}{p\tan ax+q}\right)$

27.

$\displaystyle \int\sin^{\displaystyle m}ax\cos^{\displaystyle n}ax\,dx=\left\{\... ...\int\sin^{\displaystyle m}ax\cos^{\displaystyle n-2}ax\,dx \end{array} \right.$

28.

$\displaystyle \int\displaystyle \frac{\sin^{\displaystyle m}ax}{\cos^{\displays... ...{\sin^{\displaystyle m-2}ax}{\cos^{\displaystyle n}ax}\,dx \end{array} \right.$

29.

$\displaystyle \int\displaystyle \frac{\cos^{\displaystyle m}ax}{\sin^{\displays... ...{\cos^{\displaystyle m-2}ax}{\sin^{\displaystyle n}ax}\,dx \end{array} \right.$

30.

$\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle m}ax\cos^{\displa... ...ac{dx}{\sin^{\displaystyle m-2}ax\cos^{\displaystyle n}ax} \end{array} \right.$