Evaluate the integral. (Use C for the constant of integration.)(x2 + 2x) cos(x) dx

To solve the integral of (x^2 + 2x) cos(x) dx, we can use integration by parts, which is given by the formula ∫udv = uv - ∫vdu.

Step 1: Choose u and dv Let's choose u = x^2 + 2x and dv = cos(x) dx.

Step 2: Compute du and v Then, we compute du and v. The derivative of u (du) is (2x + 2) dx and the integral of dv (v) is sin(x).

Step 3: Apply the integration by parts formula Now, we can apply the integration by parts formula: ∫udv = uv - ∫vdu.

This gives us: ∫(x^2 + 2x) cos(x) dx = (x^2 + 2x) sin(x) - ∫ sin(x) (2x + 2) dx

Step 4: Solve the remaining integral The remaining integral can be solved again by integration by parts. Let's choose u = 2x + 2 and dv = sin(x) dx. Then, du = 2 dx and v = -cos(x).

Applying the integration by parts formula again gives us: ∫ sin(x) (2x + 2) dx = -(2x + 2) cos(x) - ∫ -2 cos(x) dx

Solving the remaining integral gives us -2 sin(x).

Step 5: Combine everything So, the original integral becomes: ∫(x^2 + 2x) cos(x) dx = (x^2 + 2x) sin(x) + (2x + 2) cos(x) + 2 sin(x) + C

Where C is the constant of integration.