What is the integral of (4ln(4x^4))/(x^4) ?

The integral of (4ln(4x^4))/(x^4) is 8(-(ln(2))/(3x^3)+2(-(ln(x))/(3x^3)-1/(9x^3)))+C

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integral of (4ln(4x^4))/(x^4)

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

8 (- \frac{ln ( 2 )}{3 x ^{3}} + 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}})) + C

Solution steps

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

Simplify

ln (4 x^{4}) : 2 (2 ln (x) + ln (2))

= 4 \cdot \int \frac{2 ( ln ( 2 ) + 2 ln ( x ) )}{x ^{4}} d x

Take the constant out:

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

= 4 \cdot 2 \cdot \int \frac{ln ( 2 ) + 2 ln ( x )}{x ^{4}} d x

Expand

\frac{ln ( 2 ) + 2 ln ( x )}{x ^{4}} : \frac{ln ( 2 )}{x ^{4}} + \frac{2 ln ( x )}{x ^{4}}

= 4 \cdot 2 \cdot \int \frac{ln ( 2 )}{x ^{4}} + \frac{2 ln ( x )}{x ^{4}} 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

= 4 \cdot 2 (\int \frac{ln ( 2 )}{x ^{4}} d x + \int \frac{2 ln ( x )}{x ^{4}} d x)

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

\int \frac{2 ln ( x )}{x ^{4}} d x = 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}})

= 4 \cdot 2 (- \frac{ln ( 2 )}{3 x ^{3}} + 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}}))

Simplify

= 8 (- \frac{ln ( 2 )}{3 x ^{3}} + 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}}))

Add a constant to the solution

= 8 (- \frac{ln ( 2 )}{3 x ^{3}} + 2 (- \frac{ln ( x )}{3 x ^{3}} - \frac{1}{9 x ^{3}})) + C

\frac{d}{d x} (\frac{4 x}{2 x ^{2} + 1})

(x + 2)^{2} + (10 - 3 x)^{2}

x \to - 1 - lim (\frac{x ^{2}}{x + 1})

(1, 0), (0, 1)

\frac{d}{d x} (\frac{2 x - 5}{x})

What is the integral of (4ln(4x^4))/(x^4) ?
The integral of (4ln(4x^4))/(x^4) is 8(-(ln(2))/(3x^3)+2(-(ln(x))/(3x^3)-1/(9x^3)))+C