What is the result of ∫ (x + 1)cos(x) dx?

To find the result of the integral ∫ (x + 1)cos(x) dx, we can apply integration by parts. The integration by parts formula is given by:

∫ u dv = u v - ∫ v du

In this case, we can choose:

• u = x + 1, which gives us du = dx,

• dv = cos(x) dx, which gives us v = sin(x).

Now we can apply the integration by parts formula:

∫ (x + 1)cos(x) dx = (x + 1)sin(x) - ∫ sin(x) dx.

This shows that (x + 1)sin(x) is the product of our chosen u and v, while ∫ sin(x) dx is the integral that we need to evaluate afterwards. When we finally write it out, we add the constant of integration, C, which results in:

(x + 1)sin(x) - ∫ sin(x) dx + C.

This aligns perfectly with the first choice, where the expression correctly represents the result of integrating with parts.

It is important to note that the correct combination of the terms dictates that the sin(x) term appears with a negative sign