The integral ∫ x sinx cosx dx can be solved using the identity sinx cosx = (1/2) sin 2x, leading to the transformation of the integral into (1/2) ∫ x sin 2x dx. The solution involves integration by parts, resulting in the final expression of (-1/4)x cos 2x + (1/8) sin 2x + c. The discussion confirms the correctness of the solution without identifying any specific issues with the approach taken.
PREREQUISITESStudents studying calculus, particularly those focusing on integral calculus, as well as educators looking for examples of solving integrals involving trigonometric functions.
chapsticks said:Homework Statement
∫ x sinx cosx dx =
Homework Equations
note that sinx cosx = (1/2) sin 2x
The Attempt at a Solution
1/2) ∫ x sin 2x dx =
(1/2) [(-1/2)x cos 2x - ∫ (-1/2)cos 2x dx] =
(1/2) [(-1/2)x cos 2x +(1/2) ∫ cos 2x dx] =
(-1/4)x cos 2x +(1/4) ∫ cos 2x dx =
(-1/4)x cos 2x +(1/4)(1/2) sin 2x + c =
(-1/4)x cos 2x +(1/8) sin 2x + c =