Question

Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \begin{aligned} &\int \frac{1}{x^{2}-75} d x\\\ &\text { Partial fractions } \end{aligned} $$

Short answer

The integration of \(\int \frac{1}{x^{2}-75} dx\) is \(\frac{1}{2\sqrt{75}}(ln|x-\sqrt{75}|-ln|x+ \sqrt{75}|\) + C

Step-by-step solution

Express the rational function as a sum of partial fractions

The integral \(\int \frac{1}{x^{2}-75} dx\) can be expressed as the sum of partial fractions. The denominator can be factored as \(x^2 - 75 = (x- \sqrt{75})(x + \sqrt{75})\). Express the rational function \(\frac{1}{(x- \sqrt{75})(x + \sqrt{75})}\) as \(A/(x- \sqrt{75}) + B/(x + \sqrt{75})\).

Find the values of A and B

To determine the values of A and B, multiply both sides of the equation from step 1 by the factored denominator: \(1 = A(x + \sqrt{75}) + B(x- \sqrt{75})\). Equating the coefficients of like terms, we will find that A = 1/2\(\sqrt{75}\) and B = -1/2\(\sqrt{75}\).

Substitute A and B back into the integral

Now, substitute A and B back into the integral: \(\int \frac{1}{x^{2}-75} dx = \int \frac{1/2\sqrt{75}}{x- \sqrt{75}} dx - \int \frac{1/2\sqrt{75}}{x+ \sqrt{75}} dx\). Note that the integral of 1/x is ln|x|.

Integrate

Finally, evaluate the integral. The result is \(\frac{1}{2\sqrt{75}}(ln|x-\sqrt{75}|-ln|x+ \sqrt{75}|\) + C , where C is the constant of integration.

Key concepts

Partial Fraction Decomposition

Partial fraction decomposition is a key technique used to integrate rational functions. When dealing with a rational function, especially when the degree of the polynomial in the numerator is less than the degree in the denominator, you can express it as a sum of simpler fractions. These simpler fractions are often easier to integrate.To decompose a function into partial fractions, follow these steps:

• First, factor the polynomial in the denominator. For the integral in question, the denominator \(x^2 - 75\) can be factored into \((x - \sqrt{75})(x + \sqrt{75})\).
• Express the original function as a sum of fractions. For each factor in the denominator, assign a constant numerator. In this case, the expression \(\frac{1}{x^2 - 75}\) becomes \(\frac{A}{x - \sqrt{75}} + \frac{B}{x + \sqrt{75}}\).
• To find the constants \(A\) and \(B\), multiply through by the original denominator and simplify. Solve the resulting equation by equating coefficients for similar powers of \(x\) to find the values of \(A\) and \(B\).

Partial fraction decomposition transforms a complex fraction into simpler terms you can easily integrate. This approach is particularly effective when the integral form isn't readily apparent from the original function.

Rational Functions

Rational functions are quotients of two polynomials. They play a significant role in various branches of mathematics and are a common type of function encountered in calculus. An important aspect of rational functions is their behavior across their domain, particularly where the denominator is zero, leading to vertical asymptotes.The degree of the rational function's numerator compared to its denominator dictates integration methods. When the numerator has a smaller degree than the denominator, partially fractioning the function can simplify integration. In the example \(\int \frac{1}{x^2 - 75} dx\), the numerator is dominated by a constant (which is degree 0), making this rational function a good candidate for partial fraction decomposition.Understanding rational functions aids in predicting their behavior and how they can be simplified, offering a more direct integration approach. This can often change a challenging integral into a manageable one, with clear steps and results.

Integration Techniques

Integration techniques are methods developed to solve integrals, especially when they aren't straightforward. Common techniques include substitution, integration by parts, and partial fraction decomposition. Each technique has its scenarios where it's most effective.For the integral \(\int \frac{1}{x^2 - 75} dx\), partial fraction decomposition is the technique selected due to the nature of the rational function. After breaking the function into simpler parts, integrating each fraction separately becomes straightforward. For instance, terms like \(\int \frac{A}{x - \sqrt{75}} dx\) can be immediately integrated because the integral of \(\frac{1}{x}\) is a standard result, yielding \(ln|x|\). Here's a general approach to integration:

• Identify the form of the integral to decide which integration technique best applies.
• Apply the chosen technique, such as decomposing, substituting, or rearranging, to simplify the integral.
• Evaluate each resulting simpler integral, often using known results to expedite the process.

These techniques transform complex scenarios into smaller, actionable steps, helping you tackle a wide range of integration problems with confidence.