Question

\[\int\text{sin mx }\text{cos nx dx m }\neq n\]

Answer 1

\[\int\text{sin }\left( mx \right) \cdot \text{cos} \left( nx \right) dx\]
\[ = \frac{1}{2}\int2 \text{sin} \left( mx \right) \cdot \text{cos} \left( nx \right)dx\]
\[ = \frac{1}{2}\int\left[ \text{sin} \left( mx + nx \right) + \text{sin} \left( mx - nx \right) \right]dx \left[ \therefore \text{2 sin A }\cdot \text{cos B} = \text{sin} \left( A + B \right) + \text{sin} \left( A - B \right) \right]\]
\[ = \frac{1}{2}\left[ - \frac{\text{cos} \left( m + n \right)x}{m + n} - \frac{\text{cos} \left( m - n \right)x}{m - n} \right] + C\]