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Calculus Level 4

I = ∫ 0 ∞ x 2 ⋅ d x 2 x I={ \int _{ 0 }^{ \infty }{ \frac { { x }^{ 2 }\cdot dx }{ { 2 }^{ x } } } }

If I I can be expressed in the form A ( ln ⁡ A ) B \frac { A }{ { \left( \ln { A } \right) }^{ B } } , find A + B A+B .

This problem is original.


The answer is 5.

Jonas Katona by Jonas Katona , 22, USA

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1 solution

Do by parts by taking x^2 as the differentiable function and 2^-x as integrable function. The you would arrive at the answer 2/(ln2)^3. You could aslo make a substitution to convert it into a gamma function form. 2^-x can be written as e^(ln2^-x) = e^(-xln2). Now make the substitution xln2 = t. Then it becomes (gamma(3))/(ln2)^3. Sorry for not using latex.

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