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

The integral of 1/(sqrt(x)+\sqrt{x^3)} is 2arctan(sqrt(x))+C

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integral of 1/(sqrt(x)+\sqrt{x^3)}

\int \frac{1}{x + x ^{3}} d x

2 arctan (x) + C

Solution steps

\int \frac{1}{x + x ^{3}} d x

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

assuming

x \geq 0

= \int \frac{1}{x + x ^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{2}{1 + u ^{2}} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 2 arctan (u)

Substitute back

u = x

= 2 arctan (x)

Add a constant to the solution

= 2 arctan (x) + C

\int 3 (2 x - 1) (ln (x - 2)) d x

\frac{d}{d x} ((3 x + 1)^{- 1})

f (x) = \frac{- 5 x}{1 - x}

\frac{d}{d x} (a t^{m} + b t^{m + n})

\frac{d}{d x} (sin (x) sin (3 x))

What is the integral of 1/(sqrt(x)+\sqrt{x^3)} ?
The integral of 1/(sqrt(x)+\sqrt{x^3)} is 2arctan(sqrt(x))+C