What is the integral of xlnx - x (second integral of lnx).?

∫ x(ln(x) - 1) dx

u = x

du = dx

dv = ln(x) - 1

v = x(ln(x) - 1) - x

x²(ln(x) - 1) - x² - ∫ x(ln(x) - 1) - x dx

x²(ln(x) - 2) - ∫ x(ln(x) - 1) dx + ∫ x dx

x²(ln(x) - 2) + (x²/2) - ∫ x(ln(x) - 1) dx

x²(ln(x) - (3/2)) - ∫ x(ln(x) - 1) dx

The remaining integral is the same as the original, so we can add it to both sides and divide by 2 to give.

(x²/2)(ln(x) - (3/2))

In general, the nth integral of ln(x) is

(xⁿ/n!)(ln(x) - H(n))

where H(n) is the nth harmonic number given by ∑{i = 1, n} 1/i.

you will be able to desire to apply integration by using areas so as that (cap. S stands for ingegral) Su*dv = uv - Sv*du So for Sxln(x) you will be able to desire to compliment your u and dv, then use those to clean up for du and v. ln(x) isn't uncomplicated to combine, even with the shown fact which you may tell apart, so choose for it as your "u", consequently xdx may well be "dv" u = ln(x); dv = xdx du = (a million/x)dx; v = a million/2*x^2 So now you in simple terms plug it into the formula above Sudv = uv - Svdu = ln(x)*(x^2/2) - S(x^2/2)*(dx/x) --- pull out the a million/2, and simplify x^2/x = ln(x)*(x^2/2) - a million/2*[Sxdx] = a million/2*(x^2ln(x)) - a million/2*[a million/2*x^2] = a million/2*(x^2ln(x)) - a million/4(x^2) + C you may save simplifying from right here in case you want, yet i might probably leave it like this.