 Matematik
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Trigonometrik Değer içeren Belirli İntegral
- 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.
- \(\small \displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin px}{x}dx=\left\{ \begin{array}{ll}
\pi/2 &\hspace{.3in} p>0 \\
0&\hspace{.3in} p=0 \\
-\pi/2&\hspace{.3in} p<0
\end{array}
\right.\)
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