Matematica

Integrali con $\small \sin(ax) $ e $\small \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.: $\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle2}ax\cos ax}=\displaystyle \frac{1}{a}\ln\tan\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)\,-\,\displaystyle \frac{1}{a\sin ax}$

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

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

10.: $\displaystyle \int\displaystyle \frac{\sin^{\displaystyle2}ax}{\cos ax}\,dx=-\displaystyle \frac{\sin ax}{a}\,+\,\displaystyle \frac{1}{a}\ln\tan\left(\displaystyle \frac{ax}{2}\,+\,\displaystyle \frac{\pi}{4}\right)$

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

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

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

14.: $\displaystyle \int\displaystyle \frac{dx}{\sin ax\pm\cos ax}=\displaystyle \frac{1}{a\displaystyle \sqrt{2}}\ln\tan\left(\displaystyle \frac{ax}{2}\,\pm\,\displaystyle \frac{\pi}{8}\right)$

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.: $\displaystyle \int\displaystyle \frac{dx}{p\sin ax+q\cos ax}=\displaystyle \frac{1}{a\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}}}\ln\tan\left(\displaystyle \frac{ax+\tan^{\displaystyle-1}(q/p)}{2}\right)$

22.: $\small \displaystyle \int\displaystyle \frac{dx}{p\sin ax+q\cos ax+r}=\left\{\begin{array}{l} \displaystyle \frac{2}{a\displaystyle \sqrt{r^{\displaystyle2}-p^{\displaystyle2}-q^{\displaystyle2}}}\tan^{\displaystyle-1}\left(\displaystyle \frac{p+(r-q)\tan(ax/2)}{\displaystyle \sqrt{r^{\displaystyle2}-p^{\displaystyle2}-q^{\displaystyle2}}}\right)\\ \\ \displaystyle \frac{1}{a\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}-r^{\displaystyle2}}}\ln\left(\displaystyle \frac{p-\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}-r^{\displaystyle2}}+(r-q)\tan(ax/2)}{p+\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}-r^{\displaystyle2}}+(r-q)\tan(ax/2)}\right) \end{array} \right.$

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.: $\displaystyle \int\displaystyle \frac{dx}{p\sin ax+q\cos ax\pm\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}}}=\displaystyle \frac{-1}{a\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}}}\tan\left(\displaystyle \frac{\pi}{4}\,\mp\,\displaystyle \frac{ax+\tan^{\displaystyle-1}(q/p)}{2}\right)$

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

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

27.: $\small \displaystyle \int\sin^{\displaystyle m}ax\cos^{\displaystyle n}ax\,dx=\left\{\begin{array}{l} -\displaystyle \frac{\sin^{\displaystyle m-1}ax\cos^{\displaystyle n+1}ax}{a(m+n)}\,+\,\displaystyle \frac{m-1}{m+n}\displaystyle \int\sin^{\displaystyle m-2}ax\cos^{\displaystyle n}ax\,dx\\ \\ \displaystyle \frac{\sin^{\displaystyle m+1}ax\cos^{\displaystyle n-1}ax}{a(m+n)}\,+\,\displaystyle \frac{n-1}{m+n}\displaystyle \int\sin^{\displaystyle m}ax\cos^{\displaystyle n-2}ax\,dx \end{array} \right.$

28.: $\displaystyle \int\displaystyle \frac{\sin^{\displaystyle m}ax}{\cos^{\displaystyle n}ax}=\left\{\begin{array}{l} \displaystyle \frac{\sin^{\displaystyle m-1}ax}{a(n-1)\cos^{\displaystyle n-1}ax}\,-\,\displaystyle \frac{m-1}{n-1}\displaystyle \int\displaystyle \frac{\sin^{\displaystyle m-2}ax}{\cos^{\displaystyle n-2}ax}\,dx\\ \\ \displaystyle \frac{\sin^{\displaystyle m+1}ax}{a(n-1)\cos^{\displaystyle n-1}ax}\,-\,\displaystyle \frac{m-n+2}{n-1}\displaystyle \int\displaystyle \frac{\sin^{\displaystyle m}ax}{\cos^{\displaystyle n-2}ax}\,dx\\ \\ \displaystyle \frac{\sin^{\displaystyle m-1}ax}{a(m-n)\cos^{\displaystyle n-1}ax}\,+\,\displaystyle \frac{m-1}{m-n}\displaystyle \int\displaystyle \frac{\sin^{\displaystyle m-2}ax}{\cos^{\displaystyle n}ax}\,dx \end{array} \right.$

29.: $\displaystyle \int\displaystyle \frac{\cos^{\displaystyle m}ax}{\sin^{\displaystyle n}ax}\,dx=\left\{\begin{array}{l}, \displaystyle \frac{-\cos^{\displaystyle m-1}ax}{a(n-1)\sin^{\displaystyle n-1}ax}\,-\,\displaystyle \frac{m-1}{n-1}\displaystyle \int\displaystyle \frac{\cos^{\displaystyle m-2}ax}{\sin^{\displaystyle n-2}ax}\,dx\\ \\ \displaystyle \frac{-\cos^{\displaystyle m+1}ax}{a(n-1)\sin^{\displaystyle n-1}ax}\,-\,\displaystyle \frac{m-n+2}{n-1}\displaystyle \int\displaystyle \frac{\cos^{\displaystyle m}ax}{\sin^{\displaystyle n-2}ax}\,dx\\ \\ \displaystyle \frac{\cos^{\displaystyle m-1}ax}{a(m-n)\sin^{\displaystyle n-1}ax}\,+\,\displaystyle \frac{m-1}{m-n}\displaystyle \int\displaystyle \frac{\cos^{\displaystyle m-2}ax}{\sin^{\displaystyle n}ax}\,dx \end{array} \right.$

30.: $\small \displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle m}ax\cos^{\displaystyle n}ax}=\left\{\begin{array}{l} \displaystyle \frac{1}{a(n-1)\sin^{\displaystyle m-1}ax\cos^{\displaystyle n-1}ax}\,+\,\displaystyle \frac{m+n-2}{n-1}\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle m}ax\cos^{\displaystyle n-2}ax}\\ \\ \displaystyle \frac{-1}{a(m-1)\sin^{\displaystyle m-1}ax\cos^{\displaystyle n-1}ax}\,+\,\displaystyle \frac{m+n-2}{m-1}\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle m-2}ax\cos^{\displaystyle n}ax} \end{array} \right.$