Question

In Exercises \(1-10,\) find the indefinite integral. $$\int x e^{x} d x$$

Short answer

The indefinite integral of \(x e^{x} dx\) is \(x e^{x} - e^{x} + C\).

Step-by-step solution

Identify the Functions

Let's take the given function \(\int x e^{x} dx\) as the result of a product of two functions, namely \(u = x\) and \(dv = e^x dx\). We'll use these to start the process of integration by parts.

Find the Derivative and Integral

Find the derivative of \(u\) which gives us \(du = dx\) and the integral of \(dv\) which gives us \(v = e^x\). We now have all the parts we need to use the integration by parts formula.

Apply the Integration by Parts Formula

Use the formula for integration by parts, which is \(\int u dv = u v - \int v du\). Substituting our previously found values we get \(x e^{x} - \int e^{x} dx\).

Solve the Remaining Integral

Now solve the integral \(\int e^{x} dx\), which simplifies to \(e^{x} + C\) (where \(C\) is the constant of integration), due to the fact that \(\int e^{x} dx = e^{x}\).

Substitute the Solved Integral

Substitute this back into the expression we got from applying the integration by parts formula, giving us the final answer \(x e^{x} - e^{x} + C\).