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

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

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

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

\frac{x ^{2}}{2} - \frac{2}{3} x^{\frac{3}{2}} + x + C

Solution steps

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

Simplify

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

= \int x - x + 1 d x

Apply the Sum Rule:

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

= \int x d x - \int x d x + \int 1 d x

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

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

\int 1 d x = x

= \frac{x ^{2}}{2} - \frac{2}{3} x^{\frac{3}{2}} + x

Add a constant to the solution

= \frac{x ^{2}}{2} - \frac{2}{3} x^{\frac{3}{2}} + x + C

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