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

-substitution. Let

.
Then

, which means

Step 3: Evaluate the second integral, .
We use

-substitution. Let

.
Then

, which means

Step 4: Combine the results.

Let

b. Evaluate the following indefinite integral: (10 points)

Step 1: Use -substitution.
Notice that the derivative of the term inside the parentheses,

, is

. This suggests a

-substitution.
Let

.
Then

.
We need

, so we solve for it:

Step 2: Substitute into the integral.

Step 3: Evaluate the integral with respect to .
Using the power rule for integration (

for

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

such that its derivative simplifies and

such that it can be easily integrated. For

, we usually choose it as

.
Let

and

.
Then

and

Step 2: Apply the integration by parts formula.

Step 3: Evaluate the first term (the boundary term).

Since

and

Step 4: Evaluate the remaining integral.

Step 5: Combine the results.

To combine these, find a common denominator (9):

Final Answers:

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Calculus Integration Problems: Indefinite and Definite Integrals

Question

Answer

a. Evaluate the following indefinite integral: (5 points)

b. Evaluate the following indefinite integral: (10 points)

c. Evaluate the following definite integral: (10 points)

Final Answers:

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