What is the integral of 2/(x(x^4+25)^{1/2)} ?

The integral of 2/(x(x^4+25)^{1/2)} is -1/10 (ln|((x^4+25)^{1/2})/5+1|-ln|((x^4+25)^{1/2})/5-1|)+C

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integral of 2/(x(x^4+25)^{1/2)}

\int \frac{2}{x ( x ^{4} + 25 ) ^{\frac{1}{2}}} d x

- \frac{1}{10} ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} + 1 - ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} - 1 + C

Solution steps

\int \frac{2}{x ( x ^{4} + 25 ) ^{\frac{1}{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{1}{x ( x ^{4} + 25 ) ^{\frac{1}{2}}} d x

Apply u-substitution

: \int \frac{1}{2 ( u ^{2} - 25 )} d u

= 2 \cdot \int \frac{1}{2 ( u ^{2} - 25 )} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{2} - 25} d u

Apply Integral Substitution

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{5 ( v ^{2} - 1 )} d v

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \frac{1}{5} \cdot \int \frac{1}{v ^{2} - 1} d v

Factor

v^{2} - 1 : - (- v^{2} + 1)

= 2 \cdot \frac{1}{2} \cdot \frac{1}{5} \cdot \int \frac{1}{- ( - v ^{2} + 1 )} d v

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \frac{1}{5} (- \int \frac{1}{- v ^{2} + 1} d v)

Use the common integral:

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

= 2 \cdot \frac{1}{2} \cdot \frac{1}{5} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

= 2 \cdot \frac{1}{2} \cdot \frac{1}{5} - \frac{ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} + 1}{2} - \frac{ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} - 1}{2}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{1}{5} - \frac{ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} + 1}{2} - \frac{ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} - 1}{2} : - \frac{1}{10} ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} + 1 - ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} - 1

= - \frac{1}{10} ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} + 1 - ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} - 1

Add a constant to the solution

= - \frac{1}{10} ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} + 1 - ln \frac{( x ^{4} + 25 ) ^{\frac{1}{2}}}{5} - 1 + C

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What is the integral of 2/(x(x^4+25)^{1/2)} ?
The integral of 2/(x(x^4+25)^{1/2)} is -1/10 (ln|((x^4+25)^{1/2})/5+1|-ln|((x^4+25)^{1/2})/5-1|)+C