What is the integral from 0 to 2 of x/(x+x^2) ?

The integral from 0 to 2 of x/(x+x^2) is ln(3)

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integral from 0 to 2 of x/(x+x^2)

\int_{0}^{2} \frac{x}{x + x ^{2}} d x

ln (3)

Decimal Notation

1.09861 \dots

Solution steps

\int_{0}^{2} \frac{x}{x + x ^{2}} d x

Simplify

\frac{x}{x + x ^{2}} : \frac{1}{1 + x}

= \int_{0}^{2} \frac{1}{1 + x} d x

Apply u-substitution

: \int_{1}^{3} \frac{1}{u} d u

= \int_{1}^{3} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{1}^{3}

Compute the boundaries:

ln (3)

= ln (3)

\int \frac{9}{x ^{2}} d x

\frac{1}{10 s}

y^{^{'}} = 3 y (5 - y)

y^{^{'}} = \frac{( 2 x )}{1 + 2 y}, y (2) = 1

What is the integral from 0 to 2 of x/(x+x^2) ?
The integral from 0 to 2 of x/(x+x^2) is ln(3)