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142
∫ − π π cos n x d x = ∫ − π π sin n x d x = 0 \int_{-\pi}^{\pi}{\cos{nx}}\ \mathrm{d}x=\int_{-\pi}^{\pi}{\sin{nx}}\ \mathrm{d}x=0 ∫ − π π cos n x d x = ∫ − π π sin n x d x = 0 Derivation 143
∫ − π π cos m x sin n x d x = 0 \int_{-\pi}^{\pi}{\cos{mx}\sin{nx}}\ \mathrm{d}x=0 ∫ − π π cos m x sin n x d x = 0 Derivation 144
∫ − π π cos m x cos n x d x = { 0 ( m ≠ n ) π ( m + n ) \int_{-\pi}^{\pi}{\cos{mx}\cos{nx}}\ \mathrm{d}x=
\left\{\begin{aligned}
&0 \quad (m\neq n)\\
&\pi \quad (m+n)
\end{aligned}
\right. ∫ − π π cos m x cos n x d x = { 0 ( m = n ) π ( m + n ) Derivation 145
∫ − π π sin m x sin n x d x = { 0 ( m ≠ n ) π ( m + n ) \int_{-\pi}^{\pi}{\sin{mx}\sin{nx}}\ \mathrm{d}x=
\left\{\begin{aligned}
&0 \quad (m\neq n)\\
&\pi \quad (m+n)
\end{aligned}
\right. ∫ − π π sin m x sin n x d x = { 0 ( m = n ) π ( m + n ) Derivation 146
∫ 0 π sin m x sin n x d x = ∫ 0 π cos m x cos n x d x = { 0 ( m ≠ n ) π 2 ( m + n ) \int_{0}^{\pi}{\sin{mx}\sin{nx}}\ \mathrm{d}x= \int_{0}^{\pi}{\cos{mx}\cos{nx}}\ \mathrm{d}x=
\left\{\begin{aligned}
&0 \quad (m\neq n)\\
&\frac{\pi}{2} \quad (m+n)
\end{aligned}
\right. ∫ 0 π sin m x sin n x d x = ∫ 0 π cos m x cos n x d x = ⎩ ⎨ ⎧ 0 ( m = n ) 2 π ( m + n ) Derivation 147
∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x = { n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 4 5 ⋅ 2 3 ( n is positive odd integer greater than 1 ) n − 1 n ⋅ n − 3 n − 2 ⋅ ⋯ ⋅ 3 4 ⋅ 1 2 ⋅ π 2 ( n is positive even number ) \int_{0}^{\frac{\pi}{2}}{\sin^{n}{x}}\ \mathrm{d}x= \int_{0}^{\frac{\pi}{2}}{\cos^{n}{x}}\ \mathrm{d}x=
\left\{\begin{aligned}
&\frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \cdots \cdot \frac{4}{5}\cdot \frac{2}{3} \quad (n\text{ is positive odd integer greater than 1})\\
&\frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \cdots \cdot \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{\pi}{2} \quad (n\text{ is positive even number})
\end{aligned}
\right. ∫ 0 2 π sin n x d x = ∫ 0 2 π cos n x d x = ⎩ ⎨ ⎧ n n − 1 ⋅ n − 2 n − 3 ⋅ ⋯ ⋅ 5 4 ⋅ 3 2 ( n is positive odd integer greater than 1 ) n n − 1 ⋅ n − 2 n − 3 ⋅ ⋯ ⋅ 4 3 ⋅ 2 1 ⋅ 2 π ( n is positive even number ) Derivation