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

The integral of x(1+x)^{1/2} is 2/3 x(1+x)^{3/2}-4/15 (1+x)^{5/2}+C

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

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

\frac{2}{3} x (1 + x)^{\frac{3}{2}} - \frac{4}{15} (1 + x)^{\frac{5}{2}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{2}{3} (1 + x)^{\frac{3}{2}} d x = \frac{4}{15} (1 + x)^{\frac{5}{2}}

= \frac{2}{3} x (1 + x)^{\frac{3}{2}} - \frac{4}{15} (1 + x)^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{3} x (1 + x)^{\frac{3}{2}} - \frac{4}{15} (1 + x)^{\frac{5}{2}} + C

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What is the integral of x(1+x)^{1/2} ?
The integral of x(1+x)^{1/2} is 2/3 x(1+x)^{3/2}-4/15 (1+x)^{5/2}+C