INTEGRALS CONTAINING Cos(ax)

1.

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

2.

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

3.

$\displaystyle \int x^{\displaystyle2}\cos ax\,dx=\displaystyle \frac{2x}{a^{\di... ...displaystyle2}}{a}\,-\,\displaystyle \frac{2}{a^{\displaystyle3}}\right)\sin ax$

4.

$\displaystyle \int x^{\displaystyle3}\cos ax\,dx=\left(\displaystyle \frac{3x^{... ...isplaystyle3}}{a}\,-\,\displaystyle \frac{6x}{a^{\displaystyle3}}\right)\sin ax$

5.

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

6.

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

7.

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

8.

$\displaystyle \int\displaystyle \frac{x\,dx}{\cos ax}=\displaystyle \frac{1}{a^... ...ac{E_{\displaystyle n}(ax)^{\displaystyle2n+2}}{(2n+2)(2n)!}\,+\,\cdots\right\}$

where the constants E_n are the Euler's numbers.

9.

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

10.

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

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.

$\displaystyle \int\displaystyle \frac{dx}{\cos^{\displaystyle3}ax}=\displaystyl... ...}\ln\tan\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)$

15.

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

16.

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

17.

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

18.

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

19.

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

20.

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

21.

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

22.

$\displaystyle \int\displaystyle \frac{dx}{p+q\cos ax}=\left\{\begin{array}{l} ... ...yle \frac{1}{2}ax-\displaystyle \sqrt{(q+p)/(q-p)}}\right) \end{array} \right.$
provided $p \neq \pm q$ .

23.

$\displaystyle \int\displaystyle \frac{dx}{(p+q\cos ax)^{\displaystyle2}}=\displ... ...tyle2}-p^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{p+q\cos ax}$
provided $p \neq \pm q$ .

24.

$\displaystyle \int\displaystyle \frac{dx}{p^{\displaystyle2}+q^{\displaystyle2}... ...yle \frac{p\tan ax}{\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}}}$
provided $p \neq 0$ .

25.

$\displaystyle \int\displaystyle \frac{dx}{p^{\displaystyle2}-q^{\displaystyle2}... ...style \sqrt{q^{\displaystyle2}-p^{\displaystyle2}}}\right) \end{array} \right.$
provided $p \neq \pm q$ and $p \neq 0$ .

26.

$\displaystyle \int x^{\displaystyle m}\cos ax\,dx=\displaystyle \frac{x^{\displ... ...{m(m-1)}{a^{\displaystyle2}}\displaystyle \int x^{\displaystyle m-2}\cos ax\,dx$

27.

$\displaystyle \int\displaystyle \frac{\cos ax}{x^{\displaystyle n}}\,dx=-\displ... ...}{n-1}\displaystyle \int\displaystyle \frac{\sin ax}{x^{\displaystyle n-1}}\,dx$

28.

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

29.

$\displaystyle \int\displaystyle \frac{dx}{\cos^{\displaystyle n}ax}=\displaysty... ...{n-2}{n-1}\displaystyle \int\displaystyle \frac{dx}{\cos^{\displaystyle n-2}ax}$

30.

$\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x\,dx}{\cos^{\displays... ...playstyle \int\displaystyle \frac{x\,dx}{\cos^{\displaystyle n-2}ax}\end{array}$