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

[Tables]

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