Problem 24 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the indefinite integrals. $$ \int \frac{x^{3}-1}{x^{4}-4 x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{1}{4} \ln|x^4 - 4x| + C\).

Step 1: Identify the Substitution

We need to find a substitution that simplifies the integral. Notice that the denominator of the integrand, \(x^4 - 4x\), can be rewritten and might suggest substitution. A natural choice is to let \(u = x^4 - 4x\). This substitution will help simplify the expression.

Step 2: Determine \(du\)

Differentiate \(u = x^4 - 4x\) with respect to \(x\) to find \(du\). This gives us \(du = (4x^3 - 4)dx\). Thus, \(du = 4(x^3 - 1)dx\). Divide both sides by 4 to express \(x^3 - 1\) in terms of \(du\): \((x^3 - 1)dx = \frac{1}{4}du\).

Step 3: Substitute into the Integral

Replace \(x^3 - 1\) using \((x^3 - 1)dx = \frac{1}{4}du\) and substitute \(u = x^4 - 4x\) into the integral. The integral becomes: \[ \int \frac{x^3 - 1}{x^4 - 4x} dx = \int \frac{\frac{1}{4}du}{u} = \frac{1}{4} \int \frac{du}{u}. \]

Step 4: Integrate with respect to u

The integral \(\int \frac{du}{u}\) is a standard natural logarithm form. Therefore, \(\int \frac{du}{u} = \ln|u| + C\), where \(C\) is the integration constant. Therefore, \[ \frac{1}{4} \int \frac{du}{u} = \frac{1}{4} \ln|u| + C. \]

Step 5: Back-Substitute for x

Replace \(u\) with the original expression in terms of \(x\), which was \(u = x^4 - 4x\). Therefore, \[ \frac{1}{4} \ln|u| + C = \frac{1}{4} \ln|x^4 - 4x| + C. \]

Key Concepts

Indefinite Integrals

Indefinite integrals are mathematical expressions representing the antiderivative of a function. In simpler terms, they help us reverse differentiation. When we evaluate an indefinite integral, we want to find a function whose derivative matches the original function we are integrating. This function, plus a constant of integration, forms our indefinite integral result.
An indefinite integral generally has the form:
\( \int f(x) \, dx = F(x) + C \)
where \( F(x) \) is the antiderivative of \( f(x) \), and \( C \) is a constant, since the derivative of a constant is zero.
Indefinite integrals are crucial since they show a wider range of possible solutions and are the stepping stones for solving many problems in calculus, such as finding areas or solving differential equations.

Substitution Method

The substitution method is a handy tool for evaluating complex integrals. It simplifies an integral by transforming it into a different variable. This can turn a hard problem into a much simpler one. Also known as \( u \)-substitution, this method involves substituting part of the integrand with a new variable \( u \).
This typically happens in stages:

• Identify the part of the integrand you want to substitute, like a function inside another function.
• Let \( u \) be equal to that part and differentiate to find \( du \).
• Express the entire integral using the new variable \( u \) and \( du \).
• Integrate the new expression.
• Back-substitute the original variable into the solution.

The goal is to recognize a piece of the integrand that simplifies your work. Remember, the derivative should appear elsewhere in the integrand to make substitution viable.

Natural Logarithm Integration

The integration of the natural logarithm form is often straightforward once you set up the integral correctly using the substitution method. It involves the integral of the form \( \int \frac{1}{u} du \), which equals \( \ln|u| + C \). This form shows up frequently in calculus when dealing with rational functions or simplifying integrals through substitution.
It's essential to note the absolute value in \( \ln|u| \) and the constant \( C \) that reminds us any constant could be the solution to the integral.
Working with natural logarithm integration helps understand exponential growth and decay processes in calculus, making it a fundamental part in scenarios involving continuous compounding in finance, or the decay of radioactive substances in physics.