Question

Vxdx

Answer 1

The function is given as,

y= ∫ 0 a x x + a−x dx (1)

By using the property ∫ 0 b f( x )dx = ∫ 0 b f( b−x )dx it can be written as,

y= ∫ 0 a a−x a−x + a−( a−x ) dx = ∫ 0 a a−x a−x + x dx (2)

By adding the equation (1) and (2), we get,

2y= ∫ 0 a x x + a−x dx + ∫ 0 a a−x a−x + x dx = ∫ 0 a x + a−x x + a−x dx = ∫ 0 a 1dx 2y= [ x ] 0 a

Further simplify the above integral as,

2y=[ a−0 ] y= a 2

Thus, the solution of the integral is a 2 .