What is the integral of (xsqrt(x))/(1+x^2sqrt(x)) ?

The integral of (xsqrt(x))/(1+x^2sqrt(x)) is 2/5 ln|1+x^2sqrt(x)|+C

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integral of (xsqrt(x))/(1+x^2sqrt(x))

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

\frac{2}{5} ln 1 + x^{2} x + C

Solution steps

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

Apply u-substitution

= \int \frac{2}{5 u} d u

Take the constant out:

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

= \frac{2}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{2}{5} ln ∣ u ∣

Substitute back

u = 1 + x^{2} x

= \frac{2}{5} ln 1 + x^{2} x

Add a constant to the solution

= \frac{2}{5} ln 1 + x^{2} x + C

What is the integral of (xsqrt(x))/(1+x^2sqrt(x)) ?
The integral of (xsqrt(x))/(1+x^2sqrt(x)) is 2/5 ln|1+x^2sqrt(x)|+C