The value of integral ₀^1 ⁻¹(1 + x^2 - x) ,dx is...

Step-by-step solution

I = ₀^1 ⁻¹(1 + x^2 - x) , dx I = ₀^1 ⁻¹ ( 1 1 + x(x - 1) ) , dx I = ₀^1 ⁻¹ ( x - (x - 1) 1 + x(x - 1) ) , dx I = ₀^1 ( ⁻¹x - ⁻¹(x - 1)) , dx Using the property ₀^a f(x) , dx = ₀^a f(a - x) , dx on the second term: ₀^1 ⁻¹(x - 1) , dx = ₀^1 ⁻¹(1 - x - 1) , dx = ₀^1 ⁻¹(-x) , dx = - ₀^1 ⁻¹x , dx I = ₀^1 ⁻¹x , dx - (- ₀^1 ⁻¹x , dx ) = 2 ₀^1 ⁻¹x , dx Using integration by parts: I = 2 [ x ⁻¹x ]₀^1 - 2 ₀^1 x 1 + x^2 , dx I = 2 ( 1 4 - 0 ) - [ (1 + x^2) ]₀^1 I = 2 - 2 Answer: 2 - 2