Intergrate x^-x

Answer 1

To integrate x^(-x), we can use the technique of integration by parts. The formula for integration by parts is:

∫ u * dv = u * v - ∫ v * du,

where u and v are functions of x, and du and dv are their respective differentials.

Let's choose u = x^(-x) and dv = dx. Taking the differentials of both sides, we have du = (-x * x^(-x-1))dx and v = ∫dv = ∫dx = x.

Using the formula for integration by parts, we can rewrite the integral as:

∫ x^(-x)dx = ∫ u * dv = u * v - ∫ v * du
= x^(-x) * x - ∫ x * (-x * x^(-x-1))dx.

Simplifying this expression, we get:

∫ x^(-x)dx = x^(1-x) + ∫ x^(2-x)dx.

Now we need to integrate ∫ x^(2-x)dx. We can rewrite this as:

∫ x^(2-x)dx = ∫ (x^2 * x^(-x))dx = ∫ (x^2 * e^(-x * ln(x)))dx,

where we have used the fact that x^(-x) can be written as e^(-x * ln(x)).

This integral does not have a simple closed form solution and requires numerical methods or special functions to evaluate.

Therefore, the final result for the integral is:

∫ x^(-x)dx = x^(1-x) + ∫ x^(2-x)dx.