Matematica

Integrali con $\small \cos(ax) $

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

3.: $\displaystyle \int x^{\displaystyle2}\cos ax\,dx=\displaystyle \frac{2x}{a^{\displaystyle2}}\cos ax\,+\,\left(\displaystyle \frac{x^{\displaystyle2}}{a}\,-\,\displaystyle \frac{2}{a^{\displaystyle3}}\right)\sin ax$

4.: $\displaystyle \int x^{\displaystyle3}\cos ax\,dx=\left(\displaystyle \frac{3x^{\displaystyle2}}{a^{\displaystyle2}}\,-\,\displaystyle \frac{6}{a^{\displaystyle4}}\right)\cos ax\,+\,\left(\displaystyle \frac{x^{\displaystyle3}}{a}\,-\,\displaystyle \frac{6x}{a^{\displaystyle3}}\right)\sin ax$

5.: $\displaystyle \int\displaystyle \frac{\cos ax}{x}\,dx=\ln x\,-\,\displaystyle \frac{(ax)^{\displaystyle2}}{2\cdot 2!}\,+\,\displaystyle \frac{(ax)^{\displaystyle4}}{4\cdot 4!}\,-\,\displaystyle \frac{(ax)^{\displaystyle6}}{6\cdot 6!}\,+\,\cdots$

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

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

8.: $\displaystyle \int\displaystyle \frac{x\,dx}{\cos ax}=\displaystyle \frac{1}{a^{\displaystyle2}}\left\{\displaystyle \frac{(ax)^{\displaystyle2}}{2}\,+\,\displaystyle \frac{(ax)^{\displaystyle4}}{8}\,+\,\displaystyle \frac{5(ax)^{\displaystyle6}}{144}\,+\,\cdots\,+\,\displaystyle \frac{E_{\displaystyle n}(ax)^{\displaystyle2n+2}}{(2n+2)(2n)!}\,+\,\cdots\right\}$

9.: $\displaystyle \int\cos^{\displaystyle2}ax\,dx=\displaystyle \frac{x}{2}\,+\,\displaystyle \frac{\sin 2ax}{4a}$

10.: $\displaystyle \int x\cos^{\displaystyle2}ax\,dx=\displaystyle \frac{x^{\displaystyle2}}{4}\,+\,\displaystyle \frac{x\sin 2ax}{4a}\,+\,\displaystyle \frac{\cos 2ax}{8a^{\displaystyle2}}$

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

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

13.: $\displaystyle \int\displaystyle \frac{dx}{\cos^{\displaystyle2}ax}=\displaystyle \frac{\tan ax}{a}$

14.: $\displaystyle \int\displaystyle \frac{dx}{\cos^{\displaystyle3}ax}=\displaystyle \frac{\sin ax}{2a\cos^{\displaystyle2}ax}\,+\,\displaystyle \frac{1}{2a}\ln\tan\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)$

15.: $\displaystyle \int\cos ax\cos px\,dx=\displaystyle \frac{\sin(a-p)x}{2(a-p)}\,+\,\displaystyle \frac{\sin(a+p)x}{2(a+p)}$

16.: $\displaystyle \int\displaystyle \frac{dx}{1-\cos ax}=-\displaystyle \frac{1}{a}\cot\displaystyle \frac{ax}{2}$

17.: $\displaystyle \int\displaystyle \frac{x\,dx}{1-\cos ax}=-\displaystyle \frac{x}{a}\cot\displaystyle \frac{ax}{2}\,+\,\displaystyle \frac{2}{a^{\displaystyle2}}\ln\sin\displaystyle \frac{ax}{2}$

18.: $\displaystyle \int\displaystyle \frac{dx}{1+\cos ax}=\displaystyle \frac{1}{a}\tan\displaystyle \frac{ax}{2}$

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

20.: $\displaystyle \int\displaystyle \frac{dx}{(1-\cos ax)^{\displaystyle2}}=-\displaystyle \frac{1}{2a}\cot\displaystyle \frac{ax}{2}\,-\,\displaystyle \frac{1}{6a}\cot^{\displaystyle3}\displaystyle \frac{ax}{2}$

21.: $\displaystyle \int\displaystyle \frac{dx}{(1+\cos ax)^{\displaystyle2}}=\displaystyle \frac{1}{2a}\tan\displaystyle \frac{ax}{2}\,+\,\displaystyle \frac{1}{6a}\tan^{\displaystyle3}\displaystyle \frac{ax}{2}$

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

23.: $\displaystyle \int\displaystyle \frac{dx}{(p+q\cos ax)^{\displaystyle2}}=\displaystyle \frac{q\sin ax}{a(q^{\displaystyle2}-p^{\displaystyle2})(p+q\cos ax)}\,-\,\displaystyle \frac{p}{q^{\displaystyle2}-p^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{p+q\cos ax}$

24.: $\displaystyle \int\displaystyle \frac{dx}{p^{\displaystyle2}+q^{\displaystyle2}\cos^{\displaystyle2}ax}=\displaystyle \frac{1}{ap\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}}}\tan^{\displaystyle-1}\displaystyle \frac{p\tan ax}{\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}}}$

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

26.: $\small \displaystyle \int x^{\displaystyle m}\cos ax\,dx=\displaystyle \frac{x^{\displaystyle m}\sin ax}{a}\,+\,\displaystyle \frac{mx^{\displaystyle m-1}}{a^{\displaystyle2}}\cos ax\,-\,\displaystyle \frac{m(m-1)}{a^{\displaystyle2}}\displaystyle \int x^{\displaystyle m-2}\cos ax\,dx$

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

28.: $\displaystyle \int\cos^{\displaystyle n}ax\,dx=\displaystyle \frac{\sin ax\cos^{\displaystyle n-1}ax}{an}\,+\,\displaystyle \frac{n-1}{n}\displaystyle \int\cos^{\displaystyle n-2}ax\,dx$

29.: $\displaystyle \int\displaystyle \frac{dx}{\cos^{\displaystyle n}ax}=\displaystyle \frac{\sin ax}{a(n-1)\cos^{\displaystyle n-1}ax}\,+\,\displaystyle \frac{n-2}{n-1}\displaystyle \int\displaystyle \frac{dx}{\cos^{\displaystyle n-2}ax}$

30.: $\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x\,dx}{\cos^{\displaystyle n}ax}&=&\displaystyle \frac{x\sin ax}{a(n-1)\cos^{\displaystyle n-1}ax}\,-\,\displaystyle \frac{1}{a^{\displaystyle2}(n-1)(n-2)\cos^{\displaystyle n-2}ax}\,\\ &&\\ &&+\,\displaystyle \frac{n-2}{n-1}\displaystyle \int\displaystyle \frac{x\,dx}{\cos^{\displaystyle n-2}ax}\end{array}$