What is the integral of y^4ln(sqrt(y)) ?

The integral of y^4ln(sqrt(y)) is 1/2 (1/5 y^5ln(y)-(y^5)/(25))+C

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integral of y^4ln(sqrt(y))

\int y^{4} ln (y) d y

\frac{1}{2} (\frac{1}{5} y^{5} ln (y) - \frac{y ^{5}}{25}) + C

Solution steps

\int y^{4} ln (y) d y

Simplify

ln (y) : \frac{1}{2} ln (y)

= \int \frac{1}{2} y^{4} ln (y) d y

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int y^{4} ln (y) d y

Apply Integration By Parts

= \frac{1}{2} (\frac{1}{5} y^{5} ln (y) - \int \frac{y ^{4}}{5} d y)

\int \frac{y ^{4}}{5} d y = \frac{y ^{5}}{25}

= \frac{1}{2} (\frac{1}{5} y^{5} ln (y) - \frac{y ^{5}}{25})

Add a constant to the solution

= \frac{1}{2} (\frac{1}{5} y^{5} ln (y) - \frac{y ^{5}}{25}) + C

What is the integral of y^4ln(sqrt(y)) ?
The integral of y^4ln(sqrt(y)) is 1/2 (1/5 y^5ln(y)-(y^5)/(25))+C