What is the integral of sqrt(x+4)ln(x+4) ?

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

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integral of sqrt(x+4)ln(x+4)

\int x + 4 ln (x + 4) d x

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

Solution steps

\int x + 4 ln (x + 4) d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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