What is the integral of sqrt(x+1)ln(x+1) ?

The integral of sqrt(x+1)ln(x+1) is 2/3 ln(x+1)(x+1)^{3/2}-4/9 (x+1)^{3/2}+C

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integral of sqrt(x+1)ln(x+1)

\int x + 1 ln (x + 1) d x

\frac{2}{3} ln (x + 1) (x + 1)^{\frac{3}{2}} - \frac{4}{9} (x + 1)^{\frac{3}{2}} + C

Solution steps

\int x + 1 ln (x + 1) d x

Apply Integration By Parts

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

\int \frac{2}{3} (x + 1)^{\frac{1}{2}} d x = \frac{4}{9} (x + 1)^{\frac{3}{2}}

= \frac{2}{3} ln (x + 1) (x + 1)^{\frac{3}{2}} - \frac{4}{9} (x + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} ln (x + 1) (x + 1)^{\frac{3}{2}} - \frac{4}{9} (x + 1)^{\frac{3}{2}} + C

What is the integral of sqrt(x+1)ln(x+1) ?
The integral of sqrt(x+1)ln(x+1) is 2/3 ln(x+1)(x+1)^{3/2}-4/9 (x+1)^{3/2}+C