How do I integrate $\int\sin(x+a)\cos x \,dx $?

Question

Do I use chain rule or trigonometric relations? The final product upon which I should integrate is in the form $0.5[(\sin2x+a)+\sin a]$. so I guess trigonometric relations. But I can not find a way to convert them.

Answer 1

Note that $$\sin(x+a)\cos x=\sin x\cos a \cos x+\cos x\sin a\cos x$$ so $$\begin{align}\int\sin(x+a)\cos x\,dx&=\cos a\int\frac12\cdot2\sin x\cos x\,dx+\sin a\int\cos^2x\,dx\\&=\frac12\cos a\int\sin2x\,dx+\frac12\sin a\int(1-\cos2x)\,dx\end{align}$$ which is easy to integrate.

Answer 2

Use that $$\sin(x+a)\cos(x)=\frac{1}{2} (\sin (a+2 x)+\sin (a))$$