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
