Question

If 5f(x) + 4f (\(\frac{1}{x}\)) = \(\frac{1}{x}\)+ 3, then \(18\int_{1}^{2}\) f(x)dx is:

• A. 10 \(l\)n 3 - 6
• B. 5 \(l\)n2 - 6
• C. 10 \(l\)n 2 - 6
• D. 5 \(l\)n 2 - 3

Answer 1

The Correct Option is C

\(5f(x)+4f(\frac{1}{x})=\frac{1}{x}+3......(i)\)
Replace \(x\rightarrow \frac{1}{x}\)
\(5f(\frac{1}{x})+4f(x)=x+3.....(ii)\)
By (i) and (ii)
\(9f(x)=\frac{5}{x}-4x+3\)
\(18\int_{1}^{2}f(x)dx=\int_{1}^{2}(\frac{10}{x}-8x+6)dx\)
\(18\int_{1}^{2}f(x)dx = \) 10 ln 2 - 6

So, the correct option is (C): 10 \(l\)n 2 - 6

Answer 2

\[5f(x) + 4f\left(\frac{1}{x}\right) = \frac{1}{x} + 3\] Replacing \(x\) with \(\frac{1}{x}\): \[5f\left(\frac{1}{x}\right) + 4f(x) = x + 3\] Solving by elimination, we get: \[f(x) = \frac{5}{9x} - \frac{4x}{9} + \frac{1}{3}\] Now, evaluate: \[18 \int_{1}^{2} f(x) \, dx = 10 \ln 2 - 6\] ✅ Therefore, \(\boldsymbol{18 \int_{1}^{2} f(x) \, dx = 10 \ln 2 - 6}\).