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Calculus Integration Problems: Indefinite and Definite Integrals
Published on March 22, 2026
Calculus
Solution to three problems involving calculus techniques: basic indefinite integration, u-substitution for powers, and integration by parts for logarithms.
Question
please answer with detail
Answer
Here are the step-by-step solutions for the given integration problems.
a. Evaluate the following indefinite integral: (5 points)
Step 1: Separate the integral using the linearity property.
Step 2: Evaluate the first integral,
We use
Then
We use
Then
Step 3: Evaluate the second integral,
We use
Then
We use
Then
Step 4: Combine the results.
Let
b. Evaluate the following indefinite integral: (10 points)
Step 1: Use
Notice that the derivative of the term inside the parentheses,
Let
Then
We need
Notice that the derivative of the term inside the parentheses,
Let
Then
We need
Step 2: Substitute into the integral.
Step 3: Evaluate the integral with respect to
Using the power rule for integration (
Using the power rule for integration (
Step 4: Substitute back
c. Evaluate the following definite integral: (10 points)
Step 1: Use Integration by Parts.
The formula for integration by parts is
We choose
Let
Then
The formula for integration by parts is
We choose
Let
Then
Step 2: Apply the integration by parts formula.
Step 3: Evaluate the first term (the boundary term).
Since
Step 4: Evaluate the remaining integral.
Step 5: Combine the results.
To combine these, find a common denominator (9):
Final Answers:
a.
b.
c.