What is the integral of (6ln(2x^6))/(x^2) ?

The integral of (6ln(2x^6))/(x^2) is 6(-(ln(2))/x+6(-(ln(x))/x-1/x))+C

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integral of (6ln(2x^6))/(x^2)

\int \frac{6 ln ( 2 x ^{6} )}{x ^{2}} d x

6 (- \frac{ln ( 2 )}{x} + 6 (- \frac{ln ( x )}{x} - \frac{1}{x})) + C

Solution steps

\int \frac{6 ln ( 2 x ^{6} )}{x ^{2}} d x

Take the constant out:

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

= 6 \cdot \int \frac{ln ( 2 x ^{6} )}{x ^{2}} d x

Simplify

ln (2 x^{6}) : 6 ln (x) + ln (2)

= 6 \cdot \int \frac{ln ( 2 ) + 6 ln ( x )}{x ^{2}} d x

Expand

\frac{ln ( 2 ) + 6 ln ( x )}{x ^{2}} : \frac{ln ( 2 )}{x ^{2}} + \frac{6 ln ( x )}{x ^{2}}

= 6 \cdot \int \frac{ln ( 2 )}{x ^{2}} + \frac{6 ln ( x )}{x ^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 6 (\int \frac{ln ( 2 )}{x ^{2}} d x + \int \frac{6 ln ( x )}{x ^{2}} d x)

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

\int \frac{6 ln ( x )}{x ^{2}} d x = 6 (- \frac{ln ( x )}{x} - \frac{1}{x})

= 6 (- \frac{ln ( 2 )}{x} + 6 (- \frac{ln ( x )}{x} - \frac{1}{x}))

Add a constant to the solution

= 6 (- \frac{ln ( 2 )}{x} + 6 (- \frac{ln ( x )}{x} - \frac{1}{x})) + C

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What is the integral of (6ln(2x^6))/(x^2) ?
The integral of (6ln(2x^6))/(x^2) is 6(-(ln(2))/x+6(-(ln(x))/x-1/x))+C