What is the integral of (2xe^{x^2})/(1+e^{x^2)} ?

The integral of (2xe^{x^2})/(1+e^{x^2)} is ln|1+e^{x^2}|+C

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integral of (2xe^{x^2})/(1+e^{x^2)}

\int \frac{2 x e ^{x^{2}}}{1 + e ^{x^{2}}} d x

ln 1 + e^{x^{2}} + C

Solution steps

\int \frac{2 x e ^{x^{2}}}{1 + e ^{x^{2}}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x e ^{x^{2}}}{1 + e ^{x^{2}}} d x

Apply u-substitution

= 2 \cdot \int \frac{1}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 1 + e^{x^{2}}

= 2 \cdot \frac{1}{2} ln 1 + e^{x^{2}}

Simplify

2 \cdot \frac{1}{2} ln 1 + e^{x^{2}} : ln 1 + e^{x^{2}}

= ln 1 + e^{x^{2}}

Add a constant to the solution

= ln 1 + e^{x^{2}} + C

What is the integral of (2xe^{x^2})/(1+e^{x^2)} ?
The integral of (2xe^{x^2})/(1+e^{x^2)} is ln|1+e^{x^2}|+C