Question

please solve it
\[\int\limits_{0}^{\pi}\sin (x) \cos(nx) dx\]

Answer 1

answer is
\[-2\div (n^{2}-1) \] ,for n=even and 0 for n= odd

Answer 2

use integration by parts twice

Answer 3

alternatively use this trig identity\[\sin (u) \cos (v)=\frac 12\left[ \sin(u+v)+\sin(u-v)\right]\]
\[\sin (x) \cos (nx)=\frac 12\left[ \sin((n+1)x)+\sin((1-n)x)\right]\]

Answer 4

good