Problem 9 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{4 x^{3}+3}{x^{4}+3 x} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\frac{4 x^{3}+3}{x^{4}+3 x} dx \) is \( \ln |x^4 + 3x| + C\).

Step 1: Assign a Variable

Assign a variable \(u = x^4 + 3x \). Take the derivative of \(u\), \(du = 4x^3 + 3\ dx.\)

Step 2: Use Substitution

Substitute \(u\) and \(du\) into the original integral. The integral becomes \(\int \frac{du}{u}.\)

Step 3: Compute the Integral

Compute the integral, the antiderivative of \(\frac{1}{u}\) is \( \ln |u| \)

Step 4: Substitute Back

Replace \(u\) with \(x^4 + 3x\), so the solution becomes \( \ln |x^4 + 3x| + C\)

Key Concepts

Integration Techniques

Integration is a powerful mathematical tool that helps us find the area under curves and determine quantities like distance and volume. It's often used to reverse differentiation. There are different techniques for solving integrals, each suitable for different kinds of problems. Here are some common techniques:

• Basic Integration: This involves finding the antiderivative of a function using standard formulas.
• Substitution: This technique simplifies an integral by changing variables.
• Integration by Parts: Useful for products of functions, based on the product rule of differentiation.
• Partial Fractions: Involves breaking down complex fractions into simpler parts.

Choosing the right technique is crucial. It often involves analyzing the function to determine the most efficient method for solving it.

Substitution Method

The substitution method is a technique that makes integration simpler by introducing a new variable.
The goal is to transform the integral into a form that's easier to solve. Here's how it works:

• Identify a part of the integral to substitute with a new variable, say \( u \).
• Express \( du \) in terms of \( dx \) so you can replace it in the integral.
• Substitute both \( u \) and \( du \) into your integral, simplifying it.
• Solve the new integral, then substitute back the original variable.

In our example, we let \( u = x^4 + 3x \), turning the integral into \( \int \frac{du}{u} \). This substitution transforms the problem into something straightforward to solve.

Antiderivatives

An antiderivative is the reverse process of differentiation. It involves determining a function whose derivative is the given function.
For example, if the derivative of \( F(x) \) is \( f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \).

• The symbol for an indefinite integral represents the antiderivative of a function, expressed as \( \int f(x)\, dx = F(x) + C \).
• \( C \) is the constant of integration, indicating that there are infinitely many antiderivatives for a function.

In our problem, the antiderivative of \( \frac{1}{u} \) is \( \ln |u| \), representing the logarithmic function that gave rise to the original derivative.

Natural Logarithm

The natural logarithm is a fundamental mathematical function, represented by \( \ln(x) \). It is the inverse of the exponential function \( e^x \).
Here's why it's important in integration:

• The integral of \( \frac{1}{x} \) is \( \ln |x| \), linking it closely to antiderivatives involving rational functions.
• In many substitution problems, logarithms naturally arise when finding antiderivatives.
• The absolute value in \( \ln |x| \) ensures the logarithm is defined for negative values of \( x \).

In our exercise, the integration of \( \frac{1}{u} \) provided the result \( \ln |u| + C \). By using the natural logarithm, we easily found the solution to the integral problem.