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Integrali definiti contenenti valori trigonometrici

Questa pagina presenta esempi di integrali definiti contenenti valori trigonometrici. Questi integrali si calcolano generalmente utilizzando i limiti di integrazione e le proprietà delle funzioni trigonometriche.

1.
\(\small \displaystyle\int_{0}^{\pi}\sin mx \sin nx dx=\left\{ \begin{array}{ll} \displaystyle0\hspace{.1in} &\mbox{$m$\space ve $n$\space tamsayi ve}\;\; m \neq n \\ \pi/2 \hspace{.1in} &\mbox{$m$\space ve $n$\space tamsayi ve}\;\; m=n \end{array} \right.\)
2.
\(\small \displaystyle\int_{0}^{\pi} \cos mx \cos nx dx=\left\{ \begin{array}{ll} \displaystyle0\hspace{.1in} &\mbox{$m$\space ve $n$\space tamsayi ve}\;\; m \neq n \\ \pi/2 \hspace{.1in} &\mbox{$m$\space ve $n$\space tamsayi and}\;\; m=n \end{array} \right.\)
3.
\(\small \displaystyle\int_{0}^{\pi}\sin mx \cos nx dx=\left\{ \begin{array}{ll} \displaystyle0\hspace{.1in} &\mbox{$m$\space ve $n$\space tamsayi ve $m+n$\space tek} \\ \displaystyle2m/(m^2-n^2)\hspace{.1in} &\mbox{$m$\space ve $n$\space tamsayi ve $m+n$\space cift} \\ \end{array} \right.\)
4.
\(\small \displaystyle\int_{0}^{\pi/2} \sin^2 x dx=\int_{0}^{\pi/2}\cos^2 dx=\displaystyle \frac{\pi}{4}\)
5.
\(\small \displaystyle\int_{0}^{\pi/2}\sin^{2m} x dx=\int_{0}^{\pi/2}\cos^{2m} x dx=\displaystyle \frac{1\cdot 3\cdot 5\cdot \cdot\cdot\cdot 2m-1}{2\cdot 4\cdot 6\cdot \cdot\cdot\cdot 2m}\left(\displaystyle \frac{\pi}{2}\right)\)
6.
\(\small \displaystyle\int_{0}^{\pi/2}\sin^{2m+1}x dx=\int_{0}^{\pi/2}\cos^{2m+1}x dx=\displaystyle \frac{2\cdot 4\cdot 6\cdot\cdot\cdot 2m}{1\cdot 3\cdot 5\cdot \cdot\cdot\cdot 2m+1}\)
7.
\(\small \displaystyle\int_{0}^{\pi/2}\sin^{2p-1}x \cos^{2q-1}x dx=\displaystyle \frac{\Gamma(p)\Gamma(q)}{2\Gamma(p+q)}\)
8.
\(\small \displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin px}{x}dx=\left\{ \begin{array}{ll} \pi/2 &\hspace{.3in} p>0 \\ 0&\hspace{.3in} p=0 \\ -\pi/2&\hspace{.3in} p<0 \end{array} \right.\)