Question

Evaluate the integral ∫(x cos(2x)) dx. (Use c for the constant of integration.)

Answer 1

To evaluate the integral ∫(x cos(2x)) dx, use integration by parts. The result is (x/2) sin(2x) - (½)(−½ cos(2x)) + c.

Step-by-step explanation:

To evaluate the integral ∫(x cos(2x)) dx, we can use integration by parts. Let u = x and dv = cos(2x) dx. Applying the integration by parts formula, we find du = dx and v = ½ sin(2x). Using these values, we can now evaluate the integral:

∫(x cos(2x)) dx = uxv - ∫v du

= x(½ sin(2x)) - ∫(½ sin(2x)) dx

= (x/2) sin(2x) - (½)(−½ cos(2x)) + c

Therefore, the final result is ∫(x cos(2x)) dx = (x/2) sin(2x) - (½)(−½ cos(2x)) + c, where c is the constant of integration.