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>
To find the area of a rectangle multiply its height by its width. For a square you only need to find the length of one of the sides (as each side is the same length) and then multiply this by itself to find the area. This is the same as saying length2 or length squared.