Integration By Parts

Integration by parts applies whenever a product of dissimilar function types appears in the integrand; here the algebraic factor x meets the exponential factor e^x, two functions that no single substitution can simplify together. The method is the reverse of the product rule for differentiation, trading a hard integral for an easier remaining one. The ILATE ordering chooses u = x as the algebraic part and dv = e^x,dx as the exponential part, because algebraic functions precede exponential ones in differentiation priority. Then du = dx and v = e^x. Applying the formula \int u,dv = uv - \int v,du gives x e^x - \int e^x,dx = x e^x - e^x + C = (x-1)e^x + C. The key insight is that differentiating x removes it entirely, leaving a clean exponential to integrate. Option (x+1)e^x results from a sign error when subtracting the remaining integral. Option x e^x stops prematurely without integrating the leftover term. Option x^2 e^x/2 wrongly treats e^x as a constant and integrates x as a simple power. As a final plausibility check, differentiating (x-1)e^x by the product rule gives e^x + (x-1)e^x = x e^x, which is exactly the integrand, confirming the antiderivative holds over all real x.