What is the integral of z^nln(z) ?

The integral of z^nln(z) is 1/(n+1)z^{n+1}ln(z)-(z^{n+1})/((n+1)^2)+C

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integral of z^nln(z)

\int z^{n} ln (z) d z

\frac{1}{n + 1} z^{n + 1} ln (z) - \frac{z ^{n + 1}}{( n + 1 ) ^{2}} + C

Solution steps

\int z^{n} ln (z) d z

Apply Integration By Parts

: \frac{1}{n + 1} z^{n + 1} ln (z) - \int \frac{z ^{n}}{n + 1} d z

= \frac{1}{n + 1} z^{n + 1} ln (z) - \int \frac{z ^{n}}{n + 1} d z

\int \frac{z ^{n}}{n + 1} d z = \frac{z ^{n + 1}}{( n + 1 ) ^{2}}

= \frac{1}{n + 1} z^{n + 1} ln (z) - \frac{z ^{n + 1}}{( n + 1 ) ^{2}}

Add a constant to the solution

= \frac{1}{n + 1} z^{n + 1} ln (z) - \frac{z ^{n + 1}}{( n + 1 ) ^{2}} + C

\frac{\partial}{\partial x} (e^{x + y} sin (x - y))

\frac{d}{d x} (\frac{2}{3} (e^{- 2 x} + e^{x}))

\frac{x ^{4}}{2 - x ^{3}}

\frac{d}{d x} (47 x^{3} - 75 x)

y^{^{''}} + 7 y^{^{'}} + 10 y = 10 t e^{4 t}

What is the integral of z^nln(z) ?
The integral of z^nln(z) is 1/(n+1)z^{n+1}ln(z)-(z^{n+1})/((n+1)^2)+C