(a) Use integration by parts to find ∫ x sin(3x) dx

The question asks for integration by parts. Therefore we need to differentiate one variable and integrate the other. First we need to decide which variable is going to be which. Algebra should always be differentiated instead of trigonometric functions if possible. Therefore take:u=x, dv/dx = sin(3x). Differentiate the first term and integrate the second term to give: du/dx =1, and v = -1/3 cos(3x) . Now apply the formulae: uv - ∫ (du/dx * v) dx . This will give us: -x/3 cos(3x) - - 1/3( ∫ cos(3x) dx ) . The answer will then be: -x/3 cos(3x) + 1/9 sin(3x) + c