What is the integral of x/((x+1)^{5/2)} ?

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

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integral of x/((x+1)^{5/2)}

\int \frac{x}{( x + 1 ) ^{\frac{5}{2}}} d x

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

Solution steps

\int \frac{x}{( x + 1 ) ^{\frac{5}{2}}} d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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