Defined-a-function-f-x-such-that-f-1-x-2f-x-nx-for-m-n-gt-1-the-value-of-1-m-2n-6f-m-x-dx-is-

Question Number 83899 by john santu last updated on 07/Mar/20

Defined a function f (x) such

that f (1 - x) + 2 f (x) = nx

for m, n > 1, the value of

\int \underset{1}{\overset{m}{}} (2 n + 6 f (\frac{m}{x})) dx is \dots

Commented by john santu last updated on 07/Mar/20

replace 1 - x by x in eq 1

f (x) + 2 f (1 - x) = n (1 - x) (eq 2)

\Rightarrow 4 f (x) + 2 f (1 - x) = 2 nx

\Rightarrow f (x) + 2 f (1 - x) = n - nx

substract (1) by (2)

3 f (x) = 3 nx - n \Rightarrow f (x) = nx - \frac{n}{3}

f (\frac{m}{x}) = \frac{nm}{x} - \frac{n}{3}

\int \underset{1}{\overset{m}{}} (2 n + 6 (\frac{nm}{x} - \frac{n}{3})) dx =

\int \underset{1}{\overset{m}{}} (2 n + \frac{6 mn}{x} - 2 n) dx = m \ln {(m)}^{6 n}