Integrate | x^7 (ln x)^2 dx ( | used in place of sigma throughout question)

Start the integration by parts process

|udv = uv - |vdu

  u = (ln x)²             dv = x⁷ dx

du = 2(ln x)/x dx         v = 1/8 x⁸

= 1/8 x⁸ (ln x)² - | 1/4(ln x)x⁷ dx

= 1/8 x⁸ (ln x)² -1/4 | x⁷(ln x) dx

Repeat the integration by parts method on the integral |x⁷(ln x) dx

u=(ln x)            dv = x⁷ dx

du = 1/x dx         v = 1/8 x⁸

= 1/8 (ln x) x⁸ - 1/8 | x⁷ dx

= 1/8 (ln x) x⁸ - 1/64 x⁸

Simplify the answer (remebering to add the constant of integration).

= 1/8 x⁸ (ln x)² -1/4 (1/8 (ln x) x⁸ - 1/64 x⁸ )

= 1/8 x⁸ (ln x)² -1/32 (ln x) x⁸ + 1/256 x⁸ + C