| 1 | 1 | |||
∫sin(nπx)sin(πx)dx=− | cos(πx)sin(nπx)−∫(− | cos(πx))nπcos(nπx)dx | ||
| π | π |
| 1 | ||
∫sin(nπx)sin(πx)dx=− | cos(πx)sin(nπx)+n∫cos(nπx)cos(πx)dx | |
| π |
| 1 | ||
∫sin(nπx)sin(πx)dx=− | cos(πx)sin(nπx)+ | |
| π |
| 1 | 1 | |||
n( | sin(πx)cos(nπx)−∫( | sin(πx))(−nπsin(nπx))dx) | ||
| π | π |
| 1 | n | |||
∫sin(nπx)sin(πx)dx=− | cos(πx)sin(nπx)+ | sin(πx)cos(nπx)+n2∫sin(πx)sin(nπx)dx | ||
| π | π |
| 1 | n | |||
(1−n2)∫sin(nπx)sin(πx)dx=− | cos(πx)sin(nπx)+ | sin(πx)cos(nπx) | ||
| π | π |
| 1 | n | |||
∫sin(nπx)sin(πx)dx= | cos(πx)sin(nπx)− | sin(πx)cos(nπx) | ||
| (n2−1)π | (n2−1)π |