Question

For the following exercises, find the antiderivatives for the functions.\(\int-\frac{2 x}{x^{4}-1}\)

Short answer

The antiderivative is \(-\frac{1}{2}\ln\left| \frac{x^2 - 1}{x^2 + 1} \right| + C\)."

Step-by-step solution

Simplify the Integrand

The given function is \(-\frac{2x}{x^4 - 1}\). Notice that the denominator can be factored, as it is a difference of squares: \(x^4 - 1 = (x^2 + 1)(x^2 - 1) = (x^2 + 1)(x - 1)(x + 1)\). However, we'll use another approach that doesn't require full partial fraction decomposition.

Substitution

To solve this integral, we use substitution. Let \(u = x^2\) which means \(du = 2x \, dx\). This makes our integral \(-\int \frac{1}{u^2 - 1} \, du\). Now, factor the denominator: \(u^2 - 1 = (u - 1)(u + 1)\).

Partial Fraction Decomposition

Decompose \(\frac{1}{u^2 - 1}\) into partial fractions: \(\frac{1}{(u - 1)(u + 1)} = \frac{A}{u - 1} + \frac{B}{u + 1}\). Solve for \(A\) and \(B\) by matching coefficients: This gives \(A = \frac{1}{2}\) and \(B = -\frac{1}{2}\).

Integrate Each Term

Now, integrate each term: \(-\int \left( \frac{1}{2(u - 1)} - \frac{1}{2(u + 1)} \right) \, du\). This gives us \(-\frac{1}{2} (\ln|u - 1| - \ln|u + 1|) + C\). Simplify using the properties of logarithms: \(-\frac{1}{2}\ln\left| \frac{u - 1}{u + 1} \right| + C\).

Back Substitute

Replace \(u\) with \(x^2\) to go back to the original variable: \(-\frac{1}{2}\ln\left| \frac{x^2 - 1}{x^2 + 1} \right| + C\). This is the antiderivative of the given function.

Key concepts

Substitution Method

The substitution method is a powerful tool in integration, similar to changing variables in algebra. It aims to simplify an integral by substituting a part of it with a single variable, thus transforming it into a simpler form that is easier to evaluate.

In this particular problem, we start with the integrand \(-\frac{2x}{x^4 - 1}\). The substitution made is \(u = x^2\), which simplifies our integral by acknowledging that \(du = 2x \, dx\). This substitution is beneficial because it reduces the complexity of the variables involved, thereby transforming our original integral into \(-\int \frac{1}{u^2 - 1} \, du\).

Using substitution can be intuitive when the integrand contains composite functions or is expressed as a complicated product of terms. It also often pairs well with other techniques, such as partial fraction decomposition, to further simplify the integration process.

• Identifying useful substitutions requires practice.
• Try substituting expressions that look like derivatives of other functions within the integral.
• Always remember to convert every part of the integral into the new variable.

Partial Fraction Decomposition

After using substitution, the new integral is \(-\int \frac{1}{u^2 - 1} \, du\). To proceed, we use partial fraction decomposition. This method breaks down complex rational expressions into a sum of simpler fractions that we can integrate more easily.

Factoring the denominator \(u^2 - 1 = (u - 1)(u + 1)\) allows us to express it as \(\frac{1}{u^2 - 1} = \frac{A}{u - 1} + \frac{B}{u + 1}\). Solving for the constants \(A\) and \(B\) involves equating coefficients and results in \(A = \frac{1}{2}\) and \(B = -\frac{1}{2}\).

Partial fractions are extremely useful when dealing with rational functions as they simplify the integral into a form for which basic integration rules apply. This method is essential when an expression in the integrand is factorable.

• Ensure the degree of the numerator is less than the degree of the denominator before starting.
• If not, perform polynomial division first.
• The method works seamlessly when the denominator factors into linear terms.

Integration Techniques

Ultimately, the integration of the expression involves understanding various integration techniques combination. Each technique complements the others and is chosen based on the specific structure of the problem.

For the integral \(-\int \left( \frac{1}{2(u - 1)} - \frac{1}{2(u + 1)} \right) \, du\), we integrate by recognizing that these are standard logarithmic forms. The results become \(-\frac{1}{2} \ln|u - 1| + \frac{1}{2} \ln|u + 1| + C\). Using logarithmic properties, we further simplify to \(-\frac{1}{2}\ln\left| \frac{u - 1}{u + 1} \right| + C\).

Finally, we must return to the original variable, completing the back substitution to achieve the antiderivative \(-\frac{1}{2}\ln\left| \frac{x^2 - 1}{x^2 + 1} \right| + C\).

• Other techniques such as integration by parts or trigonometric identities might be applicable in different scenarios.
• Understanding the fundamental theorem of calculus helps you check your work.
• Practice with diverse problems to build a toolbox of strategies.