Evaluate ∫ xe2x dx. a. 1/6x2 e3x + c b. 1/2xe2x - 1/2 xe2x + c c. 1/2xe2x - 1/4 e2x + c d. 1/2x2 - 1/8 e4x + c

The formula is integral of (udv) = uv - integral of (vdu)
We use integration by parts by letting u = x and dv = (e^2x)dx
Then du = dx, and v = (1/2)(e^2x)
integral of x(e^2x)dx = (1/2)(x)(e^2x) - integral of (1/2)(e^2x)dx
= (1/2)(x)(e^2x) - (1/4)(e^2x) + C
Therefore the correct answer is C.

The answer is (1/2)xe^(2x) - (1/4)e^(2x) + C
Solution:
Since our given integrand is the product of the functions x and e^(2x), we can use the formula for integration by parts by choosing
     u = xdv/dx = e^(2x)

By differentiating, we get
     du/dx= 1
By integrating dv/dx= e^(2x), we have
     v =∫e^(2x) dx = (1/2)e^(2x)

Then we substitute these values to the integration by parts formula:
     ∫ u(dv/dx) dx = uv −∫ v(du/dx) dx ∫ x e^(2x) dx
                          = (x) (1/2)e^(2x) - ∫ ((1/2) e^(2x)) (1) dx
                          = (1/2)xe^(2x) - (1/2)∫[e^(2x)] dx
                          = (1/2)xe^(2x) - (1/2) (1/2)e^(2x) + C
where c is the constant of integration.

Therefore, our simplified answer is now
     ∫ x e^(2x) dx = (1/2)xe^(2x) - (1/4)e^(2x) + C