Question

FInd numeric value of `int_1^esqrt(lnx)dx-int_0^1e^(x^2)dx`?

Answer 1

`approx -0.207`

Explanation:

`I = underbrace(int_1^esqrt(lnx)dx)_(I_1) color(red)(-) underbrace( int_0^1e^(x^2)dx)_(I_2)`

For `I_1`:

Let: `y = sqrt(lnx)`

• `implies { (x = e^(y^2)),(dx = 2 ye^(y^2) dy):}`

`:. I_1 = int_(0)^(1) \ 2y^2e^(y^2) dy`

Switching the letter of the dummy variable `y` back to `x`.

`I = I_1 color(red)(-) I_2`

`= (I_1 + I_2) - 2 I_2`

`= int_(0)^(1) \ underbrace(2x^2e^(x^2) + e^(x^2))_(= d/dx( x e^(x^2) )) dx - 2 underbrace( int_0^1e^(x^2)dx )_(= sqrtpi/2 "erf"i(1) approx 1.463)`

`approx e - 2(1.463) approx -0.207`