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

The integral of x(1+x^2)^{5/2} is 1/7 x^2(1+x^2)^{5/2}+1/7 (x^2+1)^{5/2}+C

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

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

\frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{5}{2} x^{3} (1 + x^{2})^{\frac{3}{2}} d x = \frac{1}{14} (5 x^{2} - 2) (x^{2} + 1)^{\frac{5}{2}}

= \frac{1}{2} x^{2} (1 + x^{2})^{\frac{5}{2}} - \frac{1}{14} (5 x^{2} - 2) (x^{2} + 1)^{\frac{5}{2}}

Simplify

\frac{1}{2} x^{2} (1 + x^{2})^{\frac{5}{2}} - \frac{1}{14} (5 x^{2} - 2) (x^{2} + 1)^{\frac{5}{2}} : \frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}}

= \frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}}

Add a constant to the solution

= \frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}} + C

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What is the integral of x(1+x^2)^{5/2} ?
The integral of x(1+x^2)^{5/2} is 1/7 x^2(1+x^2)^{5/2}+1/7 (x^2+1)^{5/2}+C