determine using integration a).sigma xsinx dx and b) sigma 5theta cos 2 theta dtheta

Question

show all methods

Answer 1

a) ∫xsinxdx. Let u=x, then du=dx; and dv=sinxdx, then v=-cosx

Integrate by parts: d(uv)/dx=vdu/dx+udv/dx, so d(-xcosx)/dx=-∫cosxdx+∫xsinxdx=-sinx+∫xsinxdx.

From this ∫xsinxdx=-xcosx+sinx.

b) ∫5tcos2tdt where t represents theta.

Let u=5t, dv=cos2tdt, so du=5dt and dt=du/5; v=sin2t/2.

∫5tcos2tdt=5tsin2t/2-(5/2)∫sin2tdt=(5/2)tsin2t+(5/4)cos2t.