Integrate x*cos(x)
As there are 2 x terms in the integral we will use integration by parts. Remember; ∫u*(dv/dx)dx = uv - ∫v*(du/dx)dx (found by integrating the product rule). From xcos(x) we need to decide which x term will be u and which will be dv/dx. The reason the origional questoin is hard to integrate is due to it having 2 x terms, using the equation ∫u*(dv/dx)dx = uv - ∫v*(du/dx)dx gives us the integral ∫v*(du/dx)dx, by picking the u and dv/dx terms correctly we can ensure this integral has only one x term. We therefor want the u term to integrate to a non-x term, so, let u=x and dv/dx = cos(x).
Now we can calculate the du/dx and v terms, firstly du/dx= 1 ( using the general rule u = axb, du/dx = (a*b)xb-1 ). And secondly our v term, found by integrating dv/dx = cos(x), hence v = sin(x) (as we know sin(x) differentiates to cos(x) ).
Finally we can sub into ∫u*(dv/dx)dx = uv - ∫v*(du/dx)dx to get ∫xsinxdx = xsin(x) - ∫sin(x)*1 ( ∫sin(x)*1 simplifies to ∫sin(x) ), completing the integral ∫sin(x) = -cos(x) the equation becomes ∫xsinxdx= xsin(x) - - cos(x), which simplifies to ∫xsinxdx = xsin(x) + cos(x).
