The goal of integration by parts is to differentiate one of the terms in the product so that it becomes 1, leaving only one term to be integrated. Take a look at this problem to understand better.
Let u = x anddv = cos(3x)dx. To apply the integration by parts formula, we need v and du. du = dx. We can find v through u-substitution.
Let u_2 = 3x. Then du_2 = 3dx and dx = (du_2)/3
We can rewrite:
=int(cosu_2) * (du_2)/3
=1/3intcosu_2du_2
=1/3sinu_2
Since u_2 = 3x:
=1/3sin(3x)
Thus, v = 1/3sin(3x).
The integration by parts formula is int(udv) = uv - int(vdu).
int(xcos(3x)) dx= 1/3sin(3x) * x - int(1 * 1/3sin(3x)dx)
int(xcos3x)dx = 1/3xsin(3x) - int(1/3sin(3x)dx)
We repeat the substitution process performed above to integrate 1/3sin(3x)
Let u = 3x. Then du = 3dx and dx = 1/3du
=int1/3sinu * 1/3du
=1/9intsinu du
=-1/9cosu
=-1/9cos(3x)
The integral of the entire expression is therefore:
int(xcos3x)dx = 1/3xsin3x+ 1/9cos3x+ C, whereC is a constant.
Hopefully this helps!
