Products and integration by parts

Integral of x eˣ \int x e^x\,dx = (x - 1)e^x + C The integral of x eˣ is (x − 1)eˣ + C. See the integration by parts steps, why u = x is the right choice, a graph and examples like x e^(2x). Integral of x ln x \int x\ln x\,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C The integral of x ln x is (x²/2) ln x − x²/4 + C. See the integration by parts steps with u = ln x, a check, a graph and the xⁿ ln x formula. Integral of x sin x \int x\sin x\,dx = \sin x - x\cos x + C The integral of x sin x is sin x − x cos x + C. See the integration by parts steps, a check by differentiating, a graph and examples like x² sin x. Integral of x cos x \int x\cos x\,dx = x\sin x + \cos x + C The integral of x cos x is x sin x + cos x + C. See the integration by parts steps, a check, a graph, definite integrals and examples like x² cos x. Integral of eˣ sin x \int e^x\sin x\,dx = \frac{e^x}{2}\left(\sin x - \cos x\right) + C The integral of eˣ sin x is ½eˣ(sin x − cos x) + C. Learn the integrate-twice-and-solve trick, a check, a graph and the e^(ax) sin(bx) formula. Integral of x² eˣ \int x^2 e^x\,dx = (x^2 - 2x + 2)e^x + C The integral of x² eˣ is (x² − 2x + 2)eˣ + C. See integration by parts twice, the tabular shortcut, a check, a graph and the xⁿ eˣ pattern.