Math Problem Statement

Match each integral to its corresponding solution, given x > 2 and C is a constant.

Solution

The image you uploaded appears to contain mathematical integration problems. The task involves determining the antiderivatives (or integrals) of the given functions.

Here’s a breakdown of the image:

  1. First Equation: 4x3x42dx\int \frac{4x^3}{x^4 - 2} \, dx The result is: ln(4x8)+C\ln(4x - 8) + C (Where CC is the constant of integration).

  2. Second Equation: 1xdx\int \frac{1}{x} \, dx The result is: ln(4x48)+C\ln(4x^4 - 8) + C

  3. Third Equation: 1x2dx\int \frac{1}{x - 2} \, dx The result is: ln(4x)+C\ln(4x) + C

Each result involves logarithmic functions.

Would you like a more detailed explanation of these integrals, or do you have any further questions on this topic?

Here are five related questions you might find interesting:

  1. What is the general formula for the integral of 1x\frac{1}{x}?
  2. How do you handle integrals involving rational functions like 4x3x42\frac{4x^3}{x^4 - 2}?
  3. Can you explain why the logarithmic function appears as the result of these integrals?
  4. What is the role of the constant CC in integration?
  5. How would you compute a definite integral for these functions?

Tip: When integrating functions like 1xa\frac{1}{x - a}, the result is always a natural logarithm, lnxa+C\ln |x - a| + C.

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Math Problem Analysis

Mathematical Concepts

Integration
Logarithmic Functions
Rational Functions

Formulas

\( \int \frac{1}{x} \, dx = \ln |x| + C \)
\( \int \frac{1}{x - a} \, dx = \ln |x - a| + C \)
\( \int \frac{f'(x)}{f(x)} \, dx = \ln |f(x)| + C \)

Theorems

Logarithmic Integral Rule
Substitution Rule for Integration

Suitable Grade Level

Grades 11-12 or early college-level calculus