Problem 10 - AP Calculus with 8 Practice Tests 15th

Question

\(\int \frac{x d x}{1+4 x^{2}}=\) (A) \(\frac{1}{8} \ln \left(1+4 x^{2}\right)+C\) (B) \(\frac{1}{4} \sqrt{1+4 x^{2}}+C\) (C) \(\frac{1}{2} \ln \left|1+4 x^{2}\right|+C\) (D) \(\frac{1}{2} \tan ^{-1} 2 x+C\)

Step-by-Step Solution

Answer

The correct answer is (A) \( \frac{1}{8} \ln (1+4 x^{2})+C \).

Step 1: Identify the Integral Form

The given integral is \( \int \frac{x \, dx}{1 + 4x^2} \). This can be approached using a substitution method to simplify the expression.

Step 2: Choose a Suitable Substitution

Select a substitution that can simplify the integration process. Use \( u = 1 + 4x^2 \). This implies \( du = 8x \, dx \). To match the integral, solve for \( x \, dx \), giving \( x \, dx = \frac{1}{8} \, du \).

Step 3: Modify the Integral

Rewrite the integral in terms of \( u \):\[ \int \frac{x \, dx}{1 + 4x^2} = \int \frac{1}{8} \cdot \frac{1}{u} \, du \]

Step 4: Integrate Using the Logarithm Formula

Recognize \( \int \frac{1}{u} \, du = \ln |u| + C \). Thus, \[ \int \frac{1}{8} \cdot \frac{1}{u} \, du = \frac{1}{8} \ln |u| + C \]

Step 5: Substitution Back to Original Variable

Substitute back the original variable: \( u = 1 + 4x^2 \). Thus, the integral becomes \[ \frac{1}{8} \ln |1 + 4x^2| + C \]

Step 6: Finalize the Answer

According to the multiple-choice options provided, the answer corresponds to option (A). Therefore, the integration results in \( \frac{1}{8} \ln (1 + 4x^2) + C \), which matches option (A).

Key Concepts

Integration Techniques

In integral calculus, integration techniques are methods or strategies used to solve integrals, especially those that are not straightforward. These techniques include various methods that help simplify or manipulate an integral to make it solvable. By having a toolkit of techniques, you can tackle integrals that might seem complicated at first glance. Some of the major techniques include:

• Substitution: Replacing variables to simplify the integral.
• Integration by Parts: Used when an integral is a product of two functions.
• Partial Fraction Decomposition: Breaking down a rational function into simpler fractions.
• Trigonometric Substitution: Useful for integrals involving square roots.
• Logarithmic Integration: Where the integral involves a natural logarithm function.

Each technique has its own set of rules and is suitable for different types of integrals. Learning when to apply which technique is essential for mastering integral calculus.

Substitution Method

The substitution method is a powerful technique in integration. It is often used when an integral contains a composite function, making it hard to integrate directly. Substitution involves replacing part of the integral with another variable, usually 'u', to make the function easier to integrate. Here's how it works:

• Select a Substitution: Choose a part of the integral to substitute with a new variable 'u'. For instance, for the integral \(\int \frac{x \, dx}{1 + 4x^2}\), let \(u = 1 + 4x^2\).
• Find the Derivative: Differentiate the chosen function to find \(du\). In this example, \(du = 8x \, dx\).
• Replace and Simplify: Change the variable in the integral to 'u', making it simpler. The integral becomes \(\int \frac{1}{8} \cdot \frac{1}{u} \, du\).
• Integrate: Solve the new integral in terms of 'u'.
• Re-substitute: Substitute back to the original variable to get the answer in terms of x.

This method helps manage complex expressions and makes many integrals solvable.

Definite and Indefinite Integrals

Integrals are categorized into two main types: definite and indefinite integrals. Understanding the difference is fundamental in calculus.

• Indefinite Integrals: These don't have limits of integration and show a family of functions plus a constant C, representing an antiderivative of the function. For example, \(\int f(x) \, dx = F(x) + C\).
• Definite Integrals: These have upper and lower limits, providing a number that represents the area under the curve. The process results in \(F(b) - F(a)\), where \(F(x)\) is an antiderivative of \(f(x)\).

For this specific exercise, the integral is indefinite, as represented by the constant \(C\) in the solution \(\frac{1}{8} \ln(1+4x^2)+C\). Recognizing the type of integral is crucial for solving it correctly and understanding the type of answer expected, whether it’s a function or a numerical value.

Logarithmic Integration

Logarithmic integration is a simple yet important technique used when solving integrals that resemble a logarithmic form. This occurs when you can transform the integral into the form \(\int \frac{1}{u} \, du = \ln |u| + C\). The power of this technique lies in its ability to turn complex quotients into easily handled logarithmic expressions.

• This method is directly applied in the solution of \(\int \frac{x \, dx}{1+4x^2}\) to become \(\frac{1}{8} \ln |u| + C\).
• Logarithmic integration can often be combined with the substitution method, making it effective for a wide range of problems involving rational functions.
• Understanding when and how to apply logarithmic integration makes solving these integrals quick and ensures solutions align with typical analytic functions.

Logarithmic integration simplifies what may appear to be a daunting problem, bridging the gap between the complexity of calculus and simple algebra, ultimately showcasing the elegance of mathematical problem solving.