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

Apr 8, 2016

The Area using a numerical methods Integral Calculator
${A}_{\Delta} = \approx 0.6325074586600712$

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

#### Explanation:

Given: $f \left(x\right) = \frac{x - {x}^{2}}{\ln} \left({x}^{2} + 1\right)$

Required: Area under $f \left(x\right) \implies x : x \in \left[1 , 2\right\}$

Solution Strategy : Use the Area definite integral for $x \in \left[1 , 2\right]$

1) Definite integral: $A r e a = {\int}_{{x}_{1}}^{{x}_{2}} f \left(x\right) \mathrm{dx}$ thus
$A r e a = {\int}_{1}^{2} \frac{x - {x}^{2}}{\ln} \left({x}^{2} + 1\right) \mathrm{dx}$

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 $1 \left({x}_{2} - {x}_{1}\right)$ and height $- 1.2467 = f \left(2\right)$
Thus area is:
${A}_{\Delta} = \frac{1}{2} \left(1\right) | - 1.24267 | \approx .621335$

The Area using a numerical methods Integral Calculator
${A}_{\Delta} = \approx 0.6325074586600712$