Question

Evaluate ∫ (with 3 as upper limit and 1 as lower limit) x + ((9x)/(x^2+1)) - ((ln(x))/(7x)) dx

Answer 1

You split it up into 3 integrals
∫ (x + ((9x)/(x^2+1)) - ((ln(x))/(7x)) ) dx
= ∫x dx , easy
+ ∫ 9x/(x^2 = 1) dx , also straight forward
+ ∫ ((ln(x))/(7x)) dx <---- the hard part, I will do that one

∫ ((ln(x))/(7x)) dx = (1/7) ∫ (ln(x)/x) dx

let u = lnx
du = (1/x)dx
dx = x du

(1/7)∫ u/x xdu
= (1/7)∫ u du
= (1/7)(1/2)u^2 + c
= (ln x)^2 /14 + c
of course we are finding the definite integral from 1 to 3, so the c value will
drop out

Hint : your final answer should be appr 11, if you don't get that, post your
work so I can check it