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

The integral of (sqrt(x^3)+1)/(sqrt(x)+1) is

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integral of (sqrt(x^3)+1)/(sqrt(x)+1)

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

Solution steps

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

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

assuming

x \geq 0

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

Expand

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{x + 1} d x = 2 (x + 1 - ln x + 1)

Add a constant to the solution

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\int \frac{1}{3 y ^{2}} d y

\int 5 (x^{2} - 5 x - 2) d x

What is the integral of (sqrt(x^3)+1)/(sqrt(x)+1) ?
The integral of (sqrt(x^3)+1)/(sqrt(x)+1) is