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

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

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

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

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

Solution steps

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

Simplify

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

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

Take the constant out:

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

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

Apply Integration By Parts

: \frac{1}{5} x^{5} ln (x) - \int \frac{x ^{4}}{5} d x

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

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

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

Add a constant to the solution

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

\int \frac{x ^{3} + 4 x + 1}{x} d x

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

\frac{7 x ^{2} - 9 x + 1}{x}

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