Question

Use the suggested u to find du and rewrite the integral in terms of u. Then find the antiderivative in terms of u and, finally, rewrite your answer in terms of x. ∫cos(3x)dx,u=3x

Answer 1

To rewrite the integral in terms of u, use the substitution u = 3x. The antiderivative of cos(u) is sin(u). Therefore, the antiderivative of the integral in terms of u is (1/3)sin(u). Substitute back u = 3x to get the final answer: (1/3)sin(3x).

Step-by-step explanation:

To rewrite the integral in terms of u, we can use the substitution u = 3x. Taking the differential of u, we find du = 3dx. Rearranging this equation, we have dx = du/3. Substituting these expressions into the original integral, we get:

∫cos(3x)dx = ∫cos(u) (du/3).

To find the antiderivative in terms of u, we can use the integral of cos(u), which is sin(u). Therefore, the antiderivative of ∫cos(u)(du/3) is (1/3)sin(u).

To rewrite the answer in terms of x, we substitute back u = 3x: (1/3)sin(u) = (1/3)sin(3x).