What is the integral of (x^3ln(sqrt(x)))/3 ?

The integral of (x^3ln(sqrt(x)))/3 is 1/3 (1/8 x^4ln(x)-(x^4)/(32))+C

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integral of (x^3ln(sqrt(x)))/3

\int \frac{x ^{3} ln ( x )}{3} d x

\frac{1}{3} (\frac{1}{8} x^{4} ln (x) - \frac{x ^{4}}{32}) + C

Solution steps

\int \frac{x ^{3} ln ( x )}{3} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int x^{3} ln (x) d x

Apply Integration By Parts

: \frac{1}{4} x^{4} ln (x) - \int \frac{x ^{3}}{8} d x

= \frac{1}{3} (\frac{1}{4} x^{4} ln (x) - \int \frac{x ^{3}}{8} d x)

\int \frac{x ^{3}}{8} d x = \frac{x ^{4}}{32}

= \frac{1}{3} (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{32})

\frac{1}{4} x^{4} ln (x) = \frac{1}{8} x^{4} ln (x)

= \frac{1}{3} (\frac{1}{8} x^{4} ln (x) - \frac{x ^{4}}{32})

Add a constant to the solution

= \frac{1}{3} (\frac{1}{8} x^{4} ln (x) - \frac{x ^{4}}{32}) + C

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What is the integral of (x^3ln(sqrt(x)))/3 ?
The integral of (x^3ln(sqrt(x)))/3 is 1/3 (1/8 x^4ln(x)-(x^4)/(32))+C