A function f x , continuous on the positive real axis, has the property that for all choices of x > 0 and y > 0 , the integral ∫ x x y f t d t is independent of x (and therefore depends only on y ). If f 2 = 2 , then ∫ 1

Question

A function f x , continuous on the positive real axis, has the property that for all choices of x &#62; 0 and y &#62; 0 , the integral &#8747; x x y f t d t is independent of x (and therefore depends only on y ). If f 2 = 2 , then &#8747; 1 e f t d t is equal to

Quantrex Academy · Accepted Answer

Correct answer: C. 4

As &#8747; x x y f t d t &#160; is independent of x , so on differentiating w.r.t. x , we get f x y y - f x = 0 &#160;&#160; (since d y d x = 0 ) Substitute x = 2 , f 2 y = 2 y &#8757; f 2 = 2 f t = 4 t (Substitute y = t 2 ) &#8658; &#8747; 1 e f t d t = 4 &#8747; 1 e d t t = 4 log e t 1 e = 4 ( as log e e = 1 )

A function f x , continuous on the positive real axis, has the property that for all choices of x > 0 and y > 0 , the integral ∫ x x y f t d t is independent of x (and therefore depends only on y ). If f 2 = 2 , then ∫ 1 e f t d t is equal to

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More from Definite Integration

A function f x , continuous on the positive real axis, has the property that for all choices of x &#62; 0 and y &#62; 0 , the integral &#8747; x x y f t d t is independent of x (and therefore depends only on y ). If f 2 = 2 , then &#8747; 1 e f t d t is equal to

Options

Correct answer

Step-by-step solution

More from Definite Integration

A function f x , continuous on the positive real axis, has the property that for all choices of x > 0 and y > 0 , the integral ∫ x x y f t d t is independent of x (and therefore depends only on y ). If f 2 = 2 , then ∫ 1 e f t d t is equal to