What is the integral of x/(x^4-5x^2+3) ?

The integral of x/(x^4-5x^2+3) is -1/(4sqrt(2))(ln|(x^2)/(sqrt(2))+1|-ln|(x^2)/(sqrt(2))-1|)+C

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integral of x/(x^4-5x^2+3)

\int \frac{x}{x ^{4} - 5 x ^{2} + 3} d x

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

Solution steps

\int \frac{x}{x ^{4} - 5 x ^{2} + 3} d x

Apply u-substitution

: \int \frac{1}{4 u ( u + 3 - 5 u )} d u

= \int \frac{1}{4 u ( u + 3 - 5 u )} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u ( u + 3 - 5 u )} d u

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{u ( u + 3 - 5 u )} = u^{- \frac{1}{2}} \frac{1}{( u + 3 - 5 )}

= \frac{1}{4} \cdot \int u^{- \frac{1}{2}} \frac{1}{u + 3 - 5} d u

Simplify

u^{- \frac{1}{2}} \frac{1}{u + 3 - 5} : \frac{1}{u ( u - 2 )}

= \frac{1}{4} \cdot \int \frac{1}{u ( u - 2 )} d u

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integral Substitution

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

Factor

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

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

Use the common integral:

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

= \frac{1}{4} \cdot 2 \cdot \frac{1}{2} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of x/(x^4-5x^2+3) ?
The integral of x/(x^4-5x^2+3) is -1/(4sqrt(2))(ln|(x^2)/(sqrt(2))+1|-ln|(x^2)/(sqrt(2))-1|)+C