₀^2 |4x - 5| ,dx =

Step-by-step solution

The expression inside the absolute value is zero when 4x - 5 = 0 x = 5 4 . The interval of integration is from 0 to 2 , so we split the integral at x = 5 4 . ₀² |4x - 5| dx = ₀^ 5 4 -(4x - 5) dx + _ 5 4 ² (4x - 5) dx = ₀^ 5 4 (5 - 4x) dx + _ 5 4 ² (4x - 5) dx Evaluating the first integral: ₀^ 5 4 (5 - 4x) dx = [ 5x - 2x^2 ]₀^ 5 4 = 5 ( 5 4 ) - 2 ( 5 4 )^2 = 25 4 - 2 ( 25 16 ) = 25 4 - 25 8 = 50 8 - 25 8 = 25 8 Evaluating the second integral: _ 5 4 ² (4x - 5) dx = [ 2x^2 - 5x ]_ 5 4 ² = ( 2(2)^2 - 5(2) ) - ( 2 ( 5 4