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 x)

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

RD
Answered by Rowan D. Maths tutor

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