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$\sin(ax)$ içeren integraller

1.
$\displaystyle \int\displaystyle \frac{dx}{ax^{\displaystyle2}+bx+c}=\left\{\begin{array}{l} \displaystyle \frac{2}{\displaystyle \sqrt{4ac-b^{\displaystyle2}}}\tan^{\displaystyle-1}\displaystyle \frac{2ax+b}{\displaystyle \sqrt{4ac-b^{\displaystyle2}}} \\ \displaystyle \frac{1}{\displaystyle \sqrt{b^{\displaystyle2}-4ac}}\ln\left(\displaystyle \frac{2ax+b-\displaystyle \sqrt{b^{\displaystyle2}-4ac}}{2ax+b+\displaystyle \sqrt{b^{\displaystyle2}-4ac}}\right) \end{array} \right.$


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


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


4.
$\displaystyle \int\displaystyle \frac{\sin ax}{x}\,dx=ax\,-\,\displaystyle \frac{(ax)^{\displaystyle3}}{3\cdot 3!}\,+\,\displaystyle \frac{(ax)^{\displaystyle5}}{5\cdot 5!}\,-\,\cdots$


5.
$\displaystyle \int\displaystyle \frac{\sin ax}{x^{\displaystyle2}}\,dx=-\displaystyle \frac{\sin ax}{x}\,+\,a\displaystyle \int\displaystyle \frac{\cos ax}{x}\,dx$


6.
$\displaystyle \int\displaystyle \frac{dx}{\sin ax}=\displaystyle \frac{1}{a}\ln(\csc ax\,-\,\cot ax)=\displaystyle \frac{1}{a}\ln\tan\displaystyle \frac{ax}{2}$


7.
$\small \displaystyle \int\displaystyle \frac{x\,dx}{\sin ax}=\displaystyle \frac{1}{a^{\displaystyle2}}\left\{ax\,+\,\displaystyle \frac{(ax)^{\displaystyle3}}{18}\,+\,\displaystyle \frac{7(ax)^{\displaystyle5}}{1800}\,+\,\cdots\,+\,\displaystyle \frac{2(2^{\displaystyle2n-1}-1)B_{n}(ax)^{\displaystyle2n+1}}{(2n+1)!}\,+\,\cdots\right\}$


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


9.
$\displaystyle \int x\sin^{\displaystyle2} ax\,dx=\displaystyle \frac{x^{\displaystyle2}}{4}\,-\,\displaystyle \frac{x\sin 2ax}{4a}\,-\,\displaystyle \frac{\cos 2ax}{8a^{\displaystyle2}}$


10.
$\displaystyle \int\sin^{\displaystyle3} ax\,dx=-\displaystyle \frac{\cos ax}{a}\,+\,\displaystyle \frac{cos^{\displaystyle3}ax}{3a}$


11.
$\displaystyle \int\sin^{\displaystyle4} ax\,dx=\displaystyle \frac{3x}{8}\,-\,\displaystyle \frac{\sin 2ax}{4a}\,+\,\displaystyle \frac{\sin 4ax}{32a}$


12.
$\displaystyle \int\displaystyle \frac{dx}{\sin ^{\displaystyle2}ax}=-\displaystyle \frac{1}{a}\cot ax$


13.
$\displaystyle \int\displaystyle \frac{dx}{\sin^{\displaystyle3} ax}=-\displaystyle \frac{\cos ax}{2a\sin^{\displaystyle2}ax}\,+\,\displaystyle \frac{1}{2a}\ln\tan\displaystyle \frac{ax}{2}$


14.
$\displaystyle \int\sin px\sin qx\,dx=\displaystyle \frac{\sin(p-q)x}{2(p-q)}\,-\,\displaystyle \frac{\sin(p+q)x}{2(p+q)}$


15.
$\displaystyle \int\displaystyle \frac{dx}{1-\sin ax}=\displaystyle \frac{1}{a}\tan\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)$


16.
$\displaystyle \int\displaystyle \frac{x\,dx}{1-\sin ax}=\displaystyle \frac{x}{a}\tan\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)\,+\,\displaystyle \frac{2}{a^{\displaystyle2}}\ln\sin\left(\displaystyle \frac{\pi}{4}\,-\,\displaystyle \frac{ax}{2}\right)$


17.
$\displaystyle \int\displaystyle \frac{dx}{1+\sin ax}=-\displaystyle \frac{1}{a}\tan\left(\displaystyle \frac{\pi}{4}\,-\,\displaystyle \frac{ax}{2}\right)$


18.
$\displaystyle \int\displaystyle \frac{x\,dx}{1+\sin ax}=-\displaystyle \frac{x}{a}\tan\left(\displaystyle \frac{\pi}{4}\,-\,\displaystyle \frac{ax}{2}\right)\,+\,\displaystyle \frac{2}{a^{\displaystyle2}}\ln\sin\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)$


19.
$\displaystyle \int\displaystyle \frac{dx}{(1-\sin ax)^{\displaystyle2}}=\displaystyle \frac{1}{2a}\tan\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)\,+\,\displaystyle \frac{1}{6a}\tan^{\displaystyle3}\left(\displaystyle \frac{\pi}{4}\,+\,\displaystyle \frac{ax}{2}\right)$


20.
$\displaystyle \int\displaystyle \frac{dx}{(1+\sin ax)^{\displaystyle2}}=-\displaystyle \frac{1}{2a}\tan\left(\displaystyle \frac{\pi}{4}\,-\,\displaystyle \frac{ax}{2}\right)\,-\,\displaystyle \frac{1}{6a}\tan^{\displaystyle3}\left(\displaystyle \frac{\pi}{4}\,-\,\displaystyle \frac{ax}{2}\right)$


21.
$\displaystyle \int\displaystyle \frac{dx}{p+q\sin ax}=\left\{\begin{array}{l} \displaystyle \frac{2}{a\displaystyle \sqrt{p^{\displaystyle2}-q^{\displaystyle2}}}\tan^{\displaystyle-1}\displaystyle \frac{p\tan\displaystyle \frac{1}{2}ax+q}{\displaystyle \sqrt{p^{\displaystyle2}-q^{\displaystyle2}}}\\ \\ \displaystyle \frac{1}{a\displaystyle \sqrt{q^{\displaystyle2}-p^{\displaystyle2}}}\ln\left(\displaystyle \frac{p\tan\displaystyle \frac{1}{2}ax+q-\displaystyle \sqrt{q^{\displaystyle2}-p^{\displaystyle2}}}{p\tan\displaystyle \frac{1}{2}ax+q+\displaystyle \sqrt{q^{\displaystyle2}-p^{\displaystyle2}}}\right) \end{array} \right.$


22.
$\displaystyle \int\displaystyle \frac{dx}{(p+q\sin ax)^{\displaystyle2}}=\displaystyle \frac{q\cos ax}{a(p^{\displaystyle2}-q^{\displaystyle2})(p+q\sin ax)}\,+\,\displaystyle \frac{p}{p^{\displaystyle2}-q^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{p+q\sin ax}$


23.
$\displaystyle \int\displaystyle \frac{dx}{p^{\displaystyle2}+q^{\displaystyle2}\sin^{\displaystyle2} ax}=\displaystyle \frac{1}{ap\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}}}\tan^{\displaystyle-1}\displaystyle \frac{\displaystyle \sqrt{p^{\displaystyle2}+q^{\displaystyle2}}\tan ax}{p}$


24.
$\displaystyle \int\displaystyle \frac{dx}{p^{\displaystyle2}- q^{\displaystyle2}\sin^{\displaystyle2}ax}=\left\{\begin{array}{l} \displaystyle \frac{1}{ap\displaystyle \sqrt{p^{\displaystyle2}-q^{\displaystyle2}}}\tan^{\displaystyle-1}\displaystyle \frac{\displaystyle \sqrt{p^{\displaystyle2}-q^{\displaystyle2}}\tan ax}{p}\\ \\ \displaystyle \frac{1}{2ap\displaystyle \sqrt{q^{\displaystyle2}-p^{\displaystyle2}}}\ln\left(\displaystyle \frac{\displaystyle \sqrt{q^{\displaystyle2}-p^{\displaystyle2}}\tan ax+p}{\displaystyle \sqrt{q^{\displaystyle2}-p^{\displaystyle2}}\tan ax-p}\right) \end{array} \right.$


25.
$\small \displaystyle \int x^{\displaystyle m}\sin ax\,dx=-\displaystyle \frac{x^{\displaystyle m}\cos ax}{a}\,+\,\displaystyle \frac{mx^{\displaystyle m-1}\sin ax}{a^{\displaystyle2}}\,-\,\displaystyle \frac{m(m-1)}{a^{\displaystyle2}}\displaystyle \int\ x^{\displaystyle m-2}\sin ax\,dx$


26.
$\displaystyle \int\displaystyle \frac{\sin ax}{x^{\displaystyle n}}\,dx=-\displaystyle \frac{\sin ax}{(n-1)x^{\displaystyle n-1}}\,+\,\displaystyle \frac{a}{n-1}\displaystyle \int\displaystyle \frac{\cos ax}{x^{\displaystyle n-1}}\,dx$


27.
$\displaystyle \int\sin^{\displaystyle n}ax\,dx=-\displaystyle \frac{\sin^{\displaystyle n-1}ax\cos ax}{an}\,+\,\displaystyle \frac{n-1}{n}\displaystyle \int\sin^{\displaystyle n-2}ax\,dx$


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


29.
$\small \displaystyle \int\displaystyle \frac{x\,dx}{\sin^{\displaystyle n}ax}=\displaystyle \frac{-x\cos ax}{a(n-1)\sin^{\displaystyle n-1}ax}\,-\,\displaystyle \frac{1}{a^{\displaystyle2}(n-1)(n-2)\sin^{\displaystyle n-2}ax}\,+\,\displaystyle \frac{n-2}{n-1}\displaystyle \int\displaystyle \frac{x\,dx}{\sin^{\displaystyle n-2}ax}$


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