INTEGRALS CONTAINING Sin(ax) And Cos(ax)

1.

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

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

8.

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

9.

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

10.

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

11.

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

12.

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

13.

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

14.

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

15.

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

16.

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

17.

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

18.

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

19.

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

20.

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

21.

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

22.

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

23.

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

24.

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

25.

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

26.

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

27.

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

28.

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

29.

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

30.

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

[Tables]

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