What is the integral of 4xln(x^4) ?

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

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integral of 4xln(x^4)

\int 4 x ln (x^{4}) d x

4 (2 x^{2} ln (x) - x^{2}) + C

Solution steps

\int 4 x ln (x^{4}) d x

Take the constant out:

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

= 4 \cdot \int x ln (x^{4}) d x

Apply Integration By Parts

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

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

\int 2 x d x = x^{2}

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

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

= 4 (2 x^{2} ln (x) - x^{2})

Add a constant to the solution

= 4 (2 x^{2} ln (x) - x^{2}) + C

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