7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

Replacing $x$ by $\left( {\frac{1}{x}} \right)$ in $(i),$ we get

$2f{\rm{ }}\left( {\frac{1}{x}} \right) - 3f(x) = \frac{1}{x}$......$(ii)$

Eliminating $f{\rm{ }}\left( {\frac{1}{x}} \right)$ from $(i)$ and $(ii),$ we get

$ - 5\;f(x) = 2x + \frac{3}{x} = \frac{{2{x^2} + 3}}{3}$

==> $f(x) = - \,\left( {\frac{{2{x^2} + 3}}{{5x}}} \right)$

$\int_1^2 {f(x)dx = } $$ - \int_1^2 {\left( {\frac{{2{x^2} + 3}}{{5x}}} \right)} \;dx = - \frac{1}{5}[{x^2} + 3{\log _e}x]_1^2$

$ = - \frac{3}{5}[1 + {\log _e}2] = - \frac{3}{5}\left[ {1 + \ln 2} \right]$.