What is the integral of 2x^3(ln(x^2)) ?

The integral of 2x^3(ln(x^2)) is x^4ln(x)-(x^4)/4+C

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integral of 2x^3(ln(x^2))

\int 2 x^{3} (ln (x^{2})) d x

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

Solution steps

\int 2 x^{3} ln (x^{2}) d x

Take the constant out:

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

= 2 \cdot \int x^{3} ln (x^{2}) d x

Apply Integration By Parts

: \frac{1}{4} x^{4} ln (x^{2}) - \int \frac{x ^{3}}{2} d x

= 2 (\frac{1}{4} x^{4} ln (x^{2}) - \int \frac{x ^{3}}{2} d x)

\int \frac{x ^{3}}{2} d x = \frac{x ^{4}}{8}

= 2 (\frac{1}{4} x^{4} ln (x^{2}) - \frac{x ^{4}}{8})

Simplify

2 (\frac{1}{4} x^{4} ln (x^{2}) - \frac{x ^{4}}{8}) : x^{4} ln (x) - \frac{x ^{4}}{4}

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

Add a constant to the solution

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

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What is the integral of 2x^3(ln(x^2)) ?
The integral of 2x^3(ln(x^2)) is x^4ln(x)-(x^4)/4+C