Question

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

Answer 1

`ln(x) ((x^2)/(2)-x) - (x^2)/(4) + x + c`

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)((x^2)/(2) - x) - \int ((x^2)/(2) - x)/(x) \, dx = ln(x)((x^2)/(2) - x) \int (x)/(2)-1 \, dx =\\ln(x) ((x^2)/(2)-x) - ((x^2)/(4)-x + k) = ln(x) ((x^2)/(2)-x) - (x^2)/(4) + x + c`