Math Problem Statement

solve with steps

Solution

To solve the integral 3dxxln(4x)\int \frac{3 \, dx}{x \ln(4x)}, we can follow these steps:

Step 1: Simplify the Integral

The integral is given by: 3dxxln(4x)\int \frac{3 \, dx}{x \ln(4x)} We can factor out the constant 33 from the integral: 3dxxln(4x)3 \int \frac{dx}{x \ln(4x)}

Step 2: Substitution

Let u=ln(4x)u = \ln(4x). Now, differentiate uu with respect to xx to find dudu: dudx=14x4=1x\frac{du}{dx} = \frac{1}{4x} \cdot 4 = \frac{1}{x} So, du=dxxdu = \frac{dx}{x}.

Step 3: Rewrite the Integral in Terms of uu

Substitute uu and dudu into the integral: 3duu3 \int \frac{du}{u}

Step 4: Integrate

The integral of 1u\frac{1}{u} with respect to uu is lnu\ln|u|: 3lnu+C3 \ln|u| + C

Step 5: Substitute Back u=ln(4x)u = \ln(4x)

Replace uu with ln(4x)\ln(4x): 3lnln(4x)+C3 \ln|\ln(4x)| + C

Final Answer

The evaluated integral is: 3dxxln(4x)=3lnln(4x)+C\int \frac{3 \, dx}{x \ln(4x)} = 3 \ln|\ln(4x)| + C

Would you like further details or have any questions about this process? Here are some related questions you might find helpful:

  1. How do you approach substitution in integrals?
  2. What are the conditions under which the natural logarithm is defined?
  3. How does the constant factor influence the result of an integral?
  4. What is the significance of the absolute value in the logarithm function?
  5. Can this method be generalized to integrals of similar forms?

Tip: Always check if a substitution will simplify the integrand to a standard form before proceeding with more complex methods.

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Math Problem Analysis

Mathematical Concepts

Integration
Substitution

Formulas

Integral substitution rule

Theorems

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Suitable Grade Level

College Level