Question:
If 5f(x) + 4f (\(\frac{1}{x}\)) = \(\frac{1}{x}\)+ 3, then \(18\int_{1}^{2}\) f(x)dx is:
If 5f(x) + 4f (\(\frac{1}{x}\)) = \(\frac{1}{x}\)+ 3, then \(18\int_{1}^{2}\) f(x)dx is:
Updated On: Sep 18, 2024
10 \(l\)n 3 - 6
5 \(l\)n2 - 6
10 \(l\)n 2 - 6
5 \(l\)n 2 - 3
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The Correct Option is C
Solution and Explanation
\(5f(x)+4f(\frac{1}{x})=\frac{1}{x}+3......(i)\)
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
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Concepts Used:
Applications of Integrals
There are distinct applications of integrals, out of which some are as follows:
In Maths
Integrals are used to find:
- The center of mass (centroid) of an area having curved sides
- The area between two curves and the area under a curve
- The curve's average value
In Physics
Integrals are used to find:
- Centre of gravity
- Mass and momentum of inertia of vehicles, satellites, and a tower
- The center of mass
- The velocity and the trajectory of a satellite at the time of placing it in orbit
- Thrust