What is the integral of 6x^5ln(x) ?

The integral of 6x^5ln(x) is x^6ln(x)-(x^6)/6+C

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integral of 6x^5ln(x)

\int 6 x^{5} ln (x) d x

x^{6} ln (x) - \frac{x ^{6}}{6} + C

Solution steps

\int 6 x^{5} ln (x) d x

Take the constant out:

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

= 6 \cdot \int x^{5} ln (x) d x

Apply Integration By Parts

= 6 (\frac{1}{6} x^{6} ln (x) - \int \frac{x ^{5}}{6} d x)

\int \frac{x ^{5}}{6} d x = \frac{x ^{6}}{36}

= 6 (\frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36})

Simplify

6 (\frac{1}{6} x^{6} ln (x) - \frac{x ^{6}}{36}) : x^{6} ln (x) - \frac{x ^{6}}{6}

= x^{6} ln (x) - \frac{x ^{6}}{6}

Add a constant to the solution

= x^{6} ln (x) - \frac{x ^{6}}{6} + C

What is the integral of 6x^5ln(x) ?
The integral of 6x^5ln(x) is x^6ln(x)-(x^6)/6+C