Standard Problems in Calculus(Part 2)

Calculus Level 5

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

Let the slope of the tangent at the point (x,f(x))(x,f(x)) be m1{m}_{1} and the slope of the line joining the point and the origin be m2{m}_{2}.

Now, the ratio of log(m1+m2)\left| log({ m }_{ 1 }+{ m }_{ 2 }) \right| and log(x)\left| log(x) \right| is 2:12:1

There are exactly two functions satisfying this property let them be f1(x){f}_{1}(x) and f2(x){f}_{2}(x)

Find 1e(f1(x)+f2(x))dx\int _{ 1 }^{ e }{ ({ f }_{ 1 }(x)+{ f }_{ 2 }(x))dx }

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