how do you integrate?

Question

something like (x^2)(ln(x))?

Answer 1

Integral ( x^2 ln(x) dx )

Use integration by parts.

Let u = ln(x). dv = x^2 dx
du = (1/x) dx. v = (1/3)x^3

uv - Integral ( v du )

(1/3)x^3 ln(x) - Integral ( (1/x)(1/3)x^3 dx )

(1/3)x^3 ln(x) - (1/3) Integral ( (1/x) x^3 dx )

(1/3)x^3 ln(x) - (1/3) Integral ( x^2 dx )

(1/3)x^3 ln(x) - (1/3) (1/3)x^3 + C

(1/3)x^3 ln(x) - (1/9)x^3 + C

Answer 2

integrate by parts

integral {(x^2)(ln(x))dx}
= 1/3 * ln(x) * x^3 - integral {1/3 * x^3 * 1/x dx}
= 1/3 * x^3 * ln (x) - 1/9 * x^3 + constant ...Final answer

u can cross check by differentiating the answer

hope this helps :)

Answer 3

= (1/9) * x^3 * (3 * log x - 1)