What is the integral of (2x^5)/(4x^6-2) ?

The integral of (2x^5)/(4x^6-2) is 1/12 ln|4x^6-2|+C

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integral of (2x^5)/(4x^6-2)

\int \frac{2 x ^{5}}{4 x ^{6} - 2} d x

\frac{1}{12} ln 4 x^{6} - 2 + C

Solution steps

\int \frac{2 x ^{5}}{4 x ^{6} - 2} d x

Take the constant out:

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

= 2 \cdot \int \frac{x ^{5}}{4 x ^{6} - 2} d x

Apply u-substitution

: \int \frac{1}{24 u} d u

= 2 \cdot \int \frac{1}{24 u} d u

Take the constant out:

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

= 2 \cdot \frac{1}{24} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{24} ln ∣ u ∣

Substitute back

u = 4 x^{6} - 2

= 2 \cdot \frac{1}{24} ln 4 x^{6} - 2

Simplify

2 \cdot \frac{1}{24} ln 4 x^{6} - 2 : \frac{1}{12} ln 4 x^{6} - 2

= \frac{1}{12} ln 4 x^{6} - 2

Add a constant to the solution

= \frac{1}{12} ln 4 x^{6} - 2 + C

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What is the integral of (2x^5)/(4x^6-2) ?
The integral of (2x^5)/(4x^6-2) is 1/12 ln|4x^6-2|+C