Math Problem Statement

Use the Log Rule to find the indefinite integral. (Use C for the constant of integration.)

x

x2 + 4

dx

Solution

To solve the indefinite integral using the Log Rule, we recognize that the Log Rule states:

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

Given the integral:

xx2+4dx\int \frac{x}{x^2 + 4} \, dx

Step-by-Step Solution:

  1. Identify the numerator as the derivative of the denominator (or part of it):

    • The denominator is x2+4x^2 + 4.
    • The derivative of x2+4x^2 + 4 is 2x2x, and we have xx in the numerator.
  2. Rewrite the integral to match the Log Rule format: xx2+4dx=122xx2+4dx\int \frac{x}{x^2 + 4} \, dx = \frac{1}{2} \int \frac{2x}{x^2 + 4} \, dx

  3. Apply the Log Rule:

    • Here, f(x)=x2+4f(x) = x^2 + 4 and f(x)=2xf'(x) = 2x.
    • So, the integral becomes: 12lnx2+4+C\frac{1}{2} \ln |x^2 + 4| + C

Final Answer:

xx2+4dx=12lnx2+4+C\int \frac{x}{x^2 + 4} \, dx = \frac{1}{2} \ln |x^2 + 4| + C

Would you like a detailed explanation of any step?


5 Related Questions:

  1. How can the Log Rule be extended to definite integrals?
  2. What other functions can be solved using the Log Rule?
  3. How do you handle integrals with ln(x)\ln(x) explicitly in them?
  4. Can the Log Rule be used for higher powers in the denominator?
  5. What substitution methods simplify similar integrals?

Tip: Always check if the derivative of the denominator (or its multiple) appears in the numerator to use the Log Rule effectively!

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Logarithmic Integration
Indefinite Integrals

Formulas

Log Rule for Integration: ∫ f'(x)/f(x) dx = ln|f(x)| + C

Theorems

Logarithmic Rule for Integration

Suitable Grade Level

College Calculus or Advanced High School Calculus (Grades 11-12)