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