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

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

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

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

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

Solution steps

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

Add a constant to the solution

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

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