What is the integral of (sqrt(4+x^6))/x ?

The integral of (sqrt(4+x^6))/x is -1/3 (ln|(sqrt(x^6+4))/2+1|-ln|(sqrt(x^6+4))/2-1|-sqrt(x^6+4))+C

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integral of (sqrt(4+x^6))/x

\int \frac{4 + x ^{6}}{x} d x

- \frac{1}{3} (ln \frac{x ^{6} + 4}{2} + 1 - ln \frac{x ^{6} + 4}{2} - 1 - x^{6} + 4) + C

Solution steps

\int \frac{4 + x ^{6}}{x} d x

Apply u-substitution

: \int \frac{4 + u}{6 u} d u

= \int \frac{4 + u}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{4 + u}{u} d u

Apply u-substitution

: \int \frac{2 v ^{2}}{v ^{2} - 4} d v

= \frac{1}{6} \cdot \int \frac{2 v ^{2}}{v ^{2} - 4} d v

Take the constant out:

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

= \frac{1}{6} \cdot 2 \cdot \int \frac{v ^{2}}{v ^{2} - 4} d v

Factor

v^{2} - 4 : - (- v^{2} + 4)

= \frac{1}{6} \cdot 2 \cdot \int \frac{v ^{2}}{- ( - v ^{2} + 4 )} d v

Take the constant out:

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

= \frac{1}{6} \cdot 2 (- \int \frac{v ^{2}}{- v ^{2} + 4} d v)

Apply Integral Substitution

= \frac{1}{6} \cdot 2 (- \int \frac{2 w ^{2}}{- w ^{2} + 1} d w)

Take the constant out:

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

= \frac{1}{6} \cdot 2 (- 2 \cdot \int \frac{w ^{2}}{- w ^{2} + 1} d w)

\frac{w ^{2}}{- w ^{2} + 1} = \frac{1}{- w ^{2} + 1} - 1

= \frac{1}{6} \cdot 2 (- 2 \cdot \int \frac{1}{- w ^{2} + 1} - 1 d w)

Apply the Sum Rule:

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

= \frac{1}{6} \cdot 2 (- 2 (\int \frac{1}{- w ^{2} + 1} d w - \int 1 d w))

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

\int 1 d w = w

= \frac{1}{6} \cdot 2 (- 2 (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2} - w))

Substitute back

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

Simplify

\frac{1}{6} \cdot 2 - 2 \frac{ln \frac{4 + x ^{6}}{2} + 1}{2} - \frac{ln \frac{4 + x ^{6}}{2} - 1}{2} - \frac{4 + x ^{6}}{2} : - \frac{1}{3} (ln \frac{x ^{6} + 4}{2} + 1 - ln \frac{x ^{6} + 4}{2} - 1 - x^{6} + 4)

= - \frac{1}{3} (ln \frac{x ^{6} + 4}{2} + 1 - ln \frac{x ^{6} + 4}{2} - 1 - x^{6} + 4)

Add a constant to the solution

= - \frac{1}{3} (ln \frac{x ^{6} + 4}{2} + 1 - ln \frac{x ^{6} + 4}{2} - 1 - x^{6} + 4) + C

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What is the integral of (sqrt(4+x^6))/x ?
The integral of (sqrt(4+x^6))/x is -1/3 (ln|(sqrt(x^6+4))/2+1|-ln|(sqrt(x^6+4))/2-1|-sqrt(x^6+4))+C