If the value of definite integral ∫ 1 a x ⋅ a − [ l o g a x ] d x where a 1 , and [ ⋅ ] denotes the greatest integer, is 2 e − 1 then the value of 5 [ a ] is ____ .

Question

If the value of definite integral ∫ 1 a ​ x ⋅ a − [ l o g a x ​ ] d x where a > 1 , and [ ⋅ ] denotes the greatest integer, is 2 e − 1 ​ then the value of 5 [ a ] is ____ .

TestHub · Accepted Answer

Let lo g a x ​ = t ⇒ a t = x ⇒ d x = a t lo g e a ​ ∴ I = ln a ∫ 0 1 ​ a t ⋅ a − ( t ) ⋅ a t d t = ln a ∫ 0 1 ​ a 2 t d t = 2 a 2 − 1 ​ = 2 e − 1 ​ ⇒ a = e ​

Mathematics - Definite Integration Question with Solution | TestHub

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