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

The integral of (x^2-1)/(sqrt(x+1)) is 2/5 (x+1)^{5/2}-4/3 (x+1)^{3/2}+C

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

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

\frac{2}{5} (x + 1)^{\frac{5}{2}} - \frac{4}{3} (x + 1)^{\frac{3}{2}} + C

Solution steps

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

Expand

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

\int \frac{x ^{2}}{x + 1} d x = \frac{2}{5} (x + 1)^{\frac{5}{2}} - \frac{4}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1

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

= \frac{2}{5} (x + 1)^{\frac{5}{2}} - \frac{4}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1 - 2 x + 1

Simplify

= \frac{2}{5} (x + 1)^{\frac{5}{2}} - \frac{4}{3} (x + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (x + 1)^{\frac{5}{2}} - \frac{4}{3} (x + 1)^{\frac{3}{2}} + C

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What is the integral of (x^2-1)/(sqrt(x+1)) ?
The integral of (x^2-1)/(sqrt(x+1)) is 2/5 (x+1)^{5/2}-4/3 (x+1)^{3/2}+C