Question

Integration of e^(y/x) dy limit (y= 0 to x)

Answer 1

e^y/x dy limit [ 0 , x]
integrate wrt y

[ 1/xe^y] put limit [ 0, x]

{ 1/xe^x - 1/x }

( e^x -1)/x

Answer 2

∫(x = 0 to 1) ∫(y = 0 to x²) e^(y/x) dy dx 
= ∫(x = 0 to 1) xe^(y/x) {for y = 0 to x^2} dx 
= ∫(x = 0 to 1) x(eˣ - 1) dx 
= ∫(x = 0 to 1) (xeˣ - x) dx 
= (xeˣ - eˣ) - (1/2)x² {for x = 0 to 1}, via integration by parts 
= 1/2.