What is the integral of (sqrt(x^3))/(x(sqrt(x)+\sqrt{x^3))} ?

The integral of (sqrt(x^3))/(x(sqrt(x)+\sqrt{x^3))} is ln|1+x|+C

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integral of (sqrt(x^3))/(x(sqrt(x)+\sqrt{x^3))}

\int \frac{x ^{3}}{x ( x + x ^{3} )} d x

ln ∣ 1 + x ∣ + C

Solution steps

\int \frac{x ^{3}}{x ( x + x ^{3} )} d x

x^{3} = x^{\frac{3}{2}},

assuming

x \geq 0

= \int \frac{x ^{\frac{3}{2}}}{x ( x + x ^{3} )} d x

Simplify

\frac{x ^{\frac{3}{2}}}{x ( x + x ^{3} )} : \frac{1}{1 + x ^{2}}

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

x^{2} = (x),

assuming

x \geq 0

= \int \frac{1}{1 + x} d x

Apply u-substitution

: \int \frac{1}{u} d u

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = 1 + x

= ln ∣ 1 + x ∣

Add a constant to the solution

= ln ∣ 1 + x ∣ + C

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What is the integral of (sqrt(x^3))/(x(sqrt(x)+\sqrt{x^3))} ?
The integral of (sqrt(x^3))/(x(sqrt(x)+\sqrt{x^3))} is ln|1+x|+C