Indefinite Integrals

Step 1: Understanding the Concept:
This problem involves integration by recognizing a specific form. The integral form is a standard and very useful result derived from the product rule of differentiation. The goal is to manipulate the given integrand into this form.
Step 2: Key Formula or Approach:
The key formula to use is: We need to rewrite the given integral to match this pattern.
Step 3: Detailed Explanation:
The given integral is: First, let's simplify the integrand. The term in the denominator is equivalent to in the numerator. Now, let's split the fraction inside the integral: Now we check if this expression fits the form .
Let's try setting .
If , then its derivative is .
Substituting these into our integral, we get: This perfectly matches the standard form.
Therefore, the result of the integration is: The problem states that the integral is equal to .
By comparing our result with the given form, we can see that: Step 4: Final Answer:
The function f(x) is .

Step 1: Understanding the Concept:
This problem requires simplifying the integrand using logarithmic properties before performing the integration.
Step 2: Key Formula or Approach:
The key property to use is .
After simplification, we use the power rule for integration: .
Step 3: Detailed Explanation:
First, simplify each term in the integrand using the property .






Substitute these back into the integral:
Now, factor the numerator and the denominator:
Assuming and , we can cancel the common factor :
Now, apply the power rule for integration:
Step 4: Final Answer:
The integral is equal to }.