What is the integral of 16x^3ln(7x) ?

The integral of 16x^3ln(7x) is 4x^4ln(7x)-x^4+C

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integral of 16x^3ln(7x)

\int 16 x^{3} ln (7 x) d x

4 x^{4} ln (7 x) - x^{4} + C

Solution steps

\int 16 x^{3} ln (7 x) d x

Take the constant out:

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

= 16 \cdot \int x^{3} ln (7 x) d x

Apply Integration By Parts

: \frac{1}{4} x^{4} ln (7 x) - \int \frac{x ^{3}}{4} d x

= 16 (\frac{1}{4} x^{4} ln (7 x) - \int \frac{x ^{3}}{4} d x)

\int \frac{x ^{3}}{4} d x = \frac{x ^{4}}{16}

= 16 (\frac{1}{4} x^{4} ln (7 x) - \frac{x ^{4}}{16})

Simplify

16 (\frac{1}{4} x^{4} ln (7 x) - \frac{x ^{4}}{16}) : 4 x^{4} ln (7 x) - x^{4}

= 4 x^{4} ln (7 x) - x^{4}

Add a constant to the solution

= 4 x^{4} ln (7 x) - x^{4} + C

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What is the integral of 16x^3ln(7x) ?
The integral of 16x^3ln(7x) is 4x^4ln(7x)-x^4+C