integration by parts -
Integration of x^2 ln(x^3+2)
Integration of x^2 ln(x^3+2)
| = integration symbol
| x^2 ln(x^3 + 2)
w = x^3 + 2
dw = 3x^2
1/3 dw = x^2
1/3 | ln(w) dw
Integration by parts
| u dv = uv - | v du
u = ln(w)
dv = dw
du = 1/w dw
v = w
1/3 | ln(w) dw
= 1/3 ln(w) w - 1/3 | w 1/w dw
= 1/3 w ln(w) - 1/3 | w/w dw
= 1/3 w ln(w) - 1/3 | 1 dw
= 1/3 w ln(w) - 1/3 w + C
Substitute back in, w = x^3 + 2
= 1/3(x^3+2)ln(x^3+2) - 1/3(x^3+2) + C
This can be factored,
= 1/3 (x^3 + 2)(ln(x^3 + 2) - 1) + C