What is the integral of (2ln(4x^2))/(x^4) ?

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

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integral of (2ln(4x^2))/(x^4)

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

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

Solution steps

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

Take the constant out:

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

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

Simplify

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int - \frac{1}{3 x ^{4}} d x = \frac{1}{9 x ^{3}}

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

Simplify

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

Add a constant to the solution

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

\int_{2}^{3} \frac{3}{( x - 1 ) ( x + 2 )} d x

\frac{\partial}{\partial x} (3 x - 2 y)

\int_{1}^{6} \frac{1}{( 6 - x ) ^{\frac{2}{3}}} d x

\int \frac{3}{2} (t + 1)^{\frac{1}{2}} d t

\frac{1}{t} \frac{d y}{d t} - \frac{1}{1 + t ^{2}} = 0

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