Integral of x eˣ: Formula, Steps and Examples

The integral of xe^x is (x - 1)e^x + C. It is the standard first example of integration by parts: a power of x times an exponential.

Graph of x eˣ and its antiderivative

How to integrate x eˣ step by step

The formula for integration by parts is \int u\,dv = uv - \int v\,du.

1. Choose the parts. Let u = x and dv = e^x\,dx.
2. Find du and v. du = dx and v = e^x.
3. Apply the formula. \int xe^x\,dx = xe^x - \int e^x\,dx
4. Finish the easy integral. \int e^x\,dx = e^x, so \int xe^x\,dx = xe^x - e^x + C = (x-1)e^x + C

Why u = x? The goal is a simpler leftover integral. Differentiating x gives 1, so the x is gone after one step. The LIATE rule (Logs, Inverse trig, Algebraic, Trig, Exponential) says the same thing: algebraic comes before exponential, so x is u.

What if you choose the other way? With u = e^x and dv = x\,dx, you get \frac{x^2}{2}e^x - \int\frac{x^2}{2}e^x\,dx. The power of x went up, so the problem got worse.

Check by differentiating

The product rule gives back exactly the integrand. The derivative of x eˣ page shows the forward direction.

Definite integrals

From 0 to 1:

From −∞ to 0: as x \to -\infty, (x-1)e^x \to 0 because the exponential wins. So

The answer is negative since xe^x < 0 for x < 0.

Worked examples

Example 1: a faster exponential. \int xe^{2x}\,dx. Let u = x, dv = e^{2x}\,dx, so v = \frac{1}{2}e^{2x}:

In general, \int xe^{kx}\,dx = \left(\frac{x}{k} - \frac{1}{k^2}\right)e^{kx} + C.

Example 2: decay. With k = -1, the general formula gives \int xe^{-x}\,dx = (-x - 1)e^{-x} + C = -(x+1)e^{-x} + C. From 0 to \infty this is 0 - (-1) = 1. This is the average waiting time in the exponential distribution with rate 1.

Example 3: a linear factor. \int(3x + 2)e^x\,dx. With u = 3x + 2, du = 3\,dx:

Example 4: a higher power. For x^2e^x, use parts twice. The result is (x^2 - 2x + 2)e^x + C, shown on the integral of x² eˣ page.

Common mistakes

1. Integrating factor by factor. \int xe^x\,dx \ne \frac{x^2}{2}e^x. The integral of a product is not the product of the integrals.
2. Choosing u = e^x. The leftover integral then contains x^2, which is harder.
3. A sign slip in the last step. It is xe^x - \int e^x\,dx, so the e^x term is subtracted. Check by differentiating.
4. Forgetting \frac{1}{k} for e^{kx}. With e^{2x}, v = \frac{1}{2}e^{2x}, not e^{2x}.

For more practice, see the integration by parts guide and the integration by parts practice problems.