Question

If `int _0^1(e^x)/(1 + x)  dx = α, "then" int_0^1(e^x)/((1 + x^2))  dx` is equal to ______.

Options

• `α - 1 + e/2`

• `α + 1 - e/2`

• `α - 1 - e/2`

• `α + 1 + e/2`

Answer 1

If `int _0^1(e^x)/(1 + x)  dx = α, "then" int_0^1(e^x)/((1 + x^2))  dx` is equal to `bbunderline(α + 1 - e/2)`.

Explanation:

Given, α = `int_0^1(e^x)/((1 + x))`

α = `int_0^1(e_"II"^x(1 + x)_"I"^-1  dx`

Solve by integration by parts

α = `(1 + x)^-1 int_0^1 e^x  dx - int_0^1(d/dx (1 + x)^-1 int e^x  dx)dx`

α = `((e^x)/(1 + x))_0^1 - int(-1)/(1 + x)^2 e^x  dx`

α = `e/2 - 1 + int 1/(1 + x)^2 e^x  dx`

`α + 1 - e/2 = int 1/(1 + x)^2 e^x  dx`