Evaluate ∫ xe2x dx. 1 2 3x A./xe + C 6 B. 1/xe2x-1 / xe2x+C 22 C. 1/xe2x-1 / e2x+C 24 1 2 1 4x D./x-/e + C 28

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 = x

dv/dx = e^ (2x)

By differentiating u, 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,

∫ x e^ (2x) dx = (1/2) xe^ (2x) - (1/4) e^ (2x) + C