Question

What is the net area between `f(x)=(x-x^2)/ln(x^2+1)` in `x in[1,2]` and the x-axis?

Answer 1

The Area using a numerical methods Integral Calculator
`A_Delta= ~~0.6325074586600712`

See explanation and the estimate using the triangular area...

Explanation:

Given: `f(x)=(x-x^2)/ln(x^2+1)`

Required: Area under `f(x) => x: x in [1,2 }`

Solution Strategy : Use the Area definite integral for `x in [1,2]`

1) Definite integral: `Area=int_(x_1)^(x_2) f(x) dx` thus
`Area= int_(1)^(2)(x-x^2)/ln(x^2+1)dx`

Now this integral has be computed numerically, because it does not
have a closed form antiderivative. So let's plot and see what we can do with that. Looking at plot we see we can estimate it with the area of triangle with base `1 (x_2-x_1)` and height `-1.2467=f(2)`
Thus area is:
`A_Delta= 1/2(1)|-1.24267| ~~.621335`

The Area using a numerical methods Integral Calculator
`A_Delta= ~~0.6325074586600712`

Not bad approximation