# Calculus 2 : Integrals

## Example Questions

### Example Question #31 : Improper Integrals

Evaluate

Explanation:

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.

### Example Question #32 : Improper Integrals

Classify each improper integral as convergent or divergent.

A)

B)

C)

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

A) Diverges

B) Diverges

C) Converges

Explanation:

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:

Explanation:

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

Explanation:

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

Explanation:

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 #31 : Improper Integrals

Evaluate the following improper integral:

Explanation:

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

Explanation:

Let

### Example Question #38 : Improper Integrals

Evaluate

Explanation:

We must solve this as an improper integral.

### Example Question #921 : Integrals

Give the indefinite integral:

Explanation:

Let . Then

and

.

The integral can be rewritten as

### Example Question #922 : Integrals

Evaluate:

Explanation:

, so the integral can be rewritten as .

Substitute .

Therefore, , , and .

The bounds of integration become and .