given f(x)=1/(x(5+6log(x)), need to find ∫ f(x)dx from 1 to e.
[ note: log(x) refers to natural log, i.e. log(x) is ln(x) ]
1. Substitution of u=5+6log(x)
du = (6/x) dx
2. substitution of du and u into f(x)
∫ f(x)dx
= ∫ dx/(x(5+6log(x))
= (1/6) ∫ (6dx/x) / u
= (1/6) ∫ du/u
= (1/6) log (u) + C
= (1/6) log (5+6log(x)) + C
3. Evaluate definite integral
[ (1/6) log (5+6log(x)) ] from 1 to e
=(1/6) [ log(5+6log(e)) - log(5+6log(1))] log(e)=1, log(1)=0
=(1/6) [ log(5+6) - log(5+6(0)]
=(1/6)[log(11)-log(5)]
