Standard Problems in Calculus(Part 2)

Calculus Level 5

We have a continous and differentiblel curve \(y=f(x)\) passing through \((1,0)\)

Let the slope of the tangent at the point \((x,f(x))\) be \({m}_{1}\) and the slope of the line joining the point and the origin be \({m}_{2}\).

Now, the ratio of \(\left| log({ m }_{ 1 }+{ m }_{ 2 }) \right|\) and \(\left| log(x) \right|\) is \(2:1\)

There are exactly two functions satisfying this property let them be \({f}_{1}(x)\) and \({f}_{2}(x)\)

Find \(\int _{ 1 }^{ e }{ ({ f }_{ 1 }(x)+{ f }_{ 2 }(x))dx }\)

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