What is the integral of (x2^{x^2})/(1+2^{x^2)} ?

The integral of (x2^{x^2})/(1+2^{x^2)} is 1/(2ln(2))ln|1+2^{x^2}|+C

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integral of (x2^{x^2})/(1+2^{x^2)}

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

\frac{1}{2 ln ( 2 )} ln 1 + 2^{x^{2}} + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 ln ( 2 ) u} d u

= \int \frac{1}{2 ln ( 2 ) u} d u

Take the constant out:

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

= \frac{1}{2 ln ( 2 )} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2 ln ( 2 )} ln ∣ u ∣

Substitute back

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

= \frac{1}{2 ln ( 2 )} ln 1 + 2^{x^{2}}

Add a constant to the solution

= \frac{1}{2 ln ( 2 )} ln 1 + 2^{x^{2}} + C

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What is the integral of (x2^{x^2})/(1+2^{x^2)} ?
The integral of (x2^{x^2})/(1+2^{x^2)} is 1/(2ln(2))ln|1+2^{x^2}|+C