Math Problem Statement
5 ln(x2 − 1) dx
Solution
To solve the integral:
we can use integration by parts. The formula for integration by parts is:
- Let , so .
- Let , so .
Now, applying the formula:
Simplifying the second integral:
Next, we split the fraction:
The integral of can be expressed as a difference of logarithms:
Now we combine everything:
where is the constant of integration.
Would you like more details, or do you have any questions?
Here are 5 related questions to explore:
- How is integration by parts applied to logarithmic functions in other cases?
- Can you use substitution to solve integrals like this?
- How would the integral change if the argument of the logarithm was different, say ?
- What are the applications of logarithmic integrals in physics or engineering?
- How can partial fraction decomposition help in solving rational integrals?
Tip: When facing complex logarithmic integrals, remember that splitting the logarithmic expression into simpler terms can often simplify the integration process.
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Math Problem Analysis
Mathematical Concepts
Integration
Logarithmic Functions
Integration by Parts
Partial Fractions
Formulas
Integration by Parts: ∫u dv = uv − ∫v du
ln(x^2 − 1) = ln((x − 1)(x + 1))
∫(1/(x^2 − 1)) dx = (1/2) ln| (x − 1)/(x + 1) |
Theorems
Integration by Parts Theorem
Logarithmic Properties
Partial Fraction Decomposition
Suitable Grade Level
Undergraduate - Calculus II