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