Please help me with this calc question?
3 Answers
Have deleted answer due to an error.
Explanation:
The first integral
The second integral
Note that the two integrals have same value
note that
The two integrals are identical.
Explanation:
We seek the greater of:
# I_1 = int_a^(2a) \ 1/x \ dx # or# I_2 = int_(3a)^(6a) \ 1/x \ dx #
Where
# I(alpha,beta) = int_(alpha)^(beta) \ 1/x \ dx #
Where
# I(alpha,beta) = [ \ ln |x| \ ]_(alpha)^(beta) #
And as
# I(alpha,beta) = ln beta - ln alpha = ln (beta/alpha) #
This integral is in fact used to define the Napier logarithm and the unexpected ratio is an alternative proof of the logarithm of a product property! Given this result we now conclude that
# I_1 = ln((2a)/a) = ln 2 #
# I_2 = ln ((6a)/(3a)) =ln 2 #
Making the two integrals identical