### All Calculus 2 Resources

## Example Questions

### Example Question #31 : Improper Integrals

Evaluate

**Possible Answers:**

**Correct answer:**

Since is not an actual number, this is an improper integral. Therefore we have to rewrite it as a limit.

We replaced the improper bound, , with a new variable, t. Then the limit will allow us to evalute the integral as this new bound, t, approaches .

Now we can evaluate the integral normally.

Then we evaluate the limit.

Thus the answer is .

### Example Question #32 : Improper Integrals

Classify each improper integral as convergent or divergent.

A)

B)

C)

**Possible Answers:**

A) Diverges

B) Diverges

C) Converges

A) Converges

B) Converges

C) Diverges

A) Converges

B) Diverges

C) Converges

A) Diverges

B) Converges

C) Converges

A) Converges

B) Converges

C) Converges

**Correct answer:**

A) Diverges

B) Diverges

C) Converges

**A) **

Write the integral as a limit,

Because we know that the natural logarithm is an increasing function, it blows up as , therefore **(A) **is divergent.

**B) **

Write as a limit,

We can use a substitution to integrate,

Now write the indefinite integral in terms of .

This limit does not exist, and therefore the integral diverges.

**C)**

Recall that improper integrals of the form will converge if and will diverge if provided that .

In this case we** **have . Therefore **(C) **Converges.

### Example Question #32 : Improper Integrals

Integrate:

**Possible Answers:**

**Correct answer:**

Step 1: Integrate...

Step 2: Add up the exponent and the denominator:

Step 3: Simplify the coefficients...

Step 4: Plug in the upper limit, ...

Step 5: Plug in the lower limit, ...

Step 6: To find the value of the integral, subtract the value of step 5 from step 4...

### Example Question #34 : Improper Integrals

he Laplace Transform is an integral transform that converts functions from the time domain to the complex frequency domain . The transformation of a function into its Laplace Transform is given by:

Where , where and are constants and is the imaginary number.

Determine the Laplace Transform of

**Possible Answers:**

**Correct answer:**

Let

### Example Question #35 : Improper Integrals

he Laplace Transform is an integral transform that converts functions from the time domain to the complex frequency domain . The transformation of a function into its Laplace Transform is given by:

Where , where and are constants and is the imaginary number.

Give the Laplace Transform of .

**Possible Answers:**

**Correct answer:**

The Laplace Transform of the derivative is given by:

Using integration by parts,

Let and

The first term becomes

and the second term becomes

The Laplace Transform therefore becomes:

### Example Question #33 : Improper Integrals

Evaluate the following improper integral:

**Possible Answers:**

**Correct answer:**

To evaluate an improper integral, we first evaluate it with a finite quantity *a* in place of negative infinity:

Now, we take the limit of this quantity as a approaches negative infinity:

### Example Question #37 : Improper Integrals

The Laplace Transform is an integral transform that converts functions from the time domain to the complex frequency domain . The transformation of a function into its Laplace Transform is given by:

Where , where and are constants and is the imaginary number.

Determine the Laplace Transform of .

**Possible Answers:**

**Correct answer:**

Let

### Example Question #38 : Improper Integrals

Evaluate

**Possible Answers:**

**Correct answer:**

We must solve this as an improper integral.

### Example Question #921 : Integrals

Give the indefinite integral:

**Possible Answers:**

**Correct answer:**

Let . Then

and

.

The integral can be rewritten as

### Example Question #922 : Integrals

Evaluate:

**Possible Answers:**

**Correct answer:**

, so the integral can be rewritten as .

Substitute .

Therefore, , , and .

The bounds of integration become and .

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