Question

Find the integral, using techniques from this or the previous chapter. ∫(x-1)ln |x| dx.

Answer 1

Answer:

$ln(x) (\frac{x^2}{2}-x) - \frac{x^2}{4} + x + c$

Step-by-step explanation:

We will use the parts method of integration. ln(x) can be derivated while we integrate the polynomium x-1.

• The derivate of ln(x) is 1/x
• The integrate of x-1 is x²/2 -x

Thus,

$\int (x-1)ln(x) \, dx = ln(x)(\frac{x^2}{2} - x) - \int \frac{\frac{x^2}{2} - x}{x} \, dx = ln(x)(\frac{x^2}{2} - x) \int \frac{x}{2}-1 \, dx =\\ln(x) (\frac{x^2}{2}-x) - (\frac{x^2}{4}-x + k) = ln(x) (\frac{x^2}{2}-x) - \frac{x^2}{4} + x + c$