help [[integrals]]??

Question

Answer 1

(10)

Step 1:

The integral is .

Continuity Implies Integrability Theorem:

If a function is continuous on the closed interval , then is integrable on .

exists.

Consider the function .

The function is continuous on the interval .

is exists.

Step 2:

Evaluation of integral:.

.

.

Solution:

.

Answer 2

(9)

Step 1:

The integral is .

Continuity Implies Integrability Theorem:

If a function is continuous on the closed interval , then is integrable on .

exists.

Consider .

The function is a linear function.

The function is continuous at every real number.

The function is continuous on the interval .

is exists.

Step 2:

Evaluation of integral:.

Apply sum and difference rule in integration: .

.

.

Solution:

.

Answer 3

(11)

Step 1:

The integral is .

Continuity Implies Integrability Theorem:

If a function is continuous on the closed interval , then is integrable on .

exists.

Consider the function .

Graph the function: .

Observe the graph:

The function is continuous on the interval .

is exists.

Comments

Contd...

Step 2:

Evaluation of integral:.

Apply special integration formula: .

In this case and .

.

.

Solution:

.