Calculus 2 : Integrals

Example Questions

Example Question #672 : Finding Integrals

Evaluate the integral:

Explanation:

To evaluate the integral, we perform the following substitution:

The derivative was found using the following rule:

Now, we rewrite the integrand and integrate:

The integral was performed using the following rule:

Finally, replace u with our original x term:

Example Question #673 : Finding 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 complex frequency function   is given by:

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

Evaluate the Laplace Transform of the function  at time . Suppose that  when

Explanation:

The Laplace Transform will be given by:

Since  when , we can change the integral to:

This is because when you change the lower bound of the integral, the exponent will only exist for values for which  is defined.

Let

This changes our integral to:

We can now move the  term out of the integral, which will give us:

Example Question #674 : Finding Integrals

Evaluate the following integral using substitution:

Explanation:

To evaluate this integral, we first make the following substitution:

Differentiating this expression, we get:

Now, we can rewrite the original integral with our substitution and solve:

Finally, we have to replace with our earlier definition:

Example Question #675 : Finding Integrals

In exponentially decaying systems, often times the solutions to differential equations take on the form of an integral called Duhamel's Integral. This is given by:

Where  is a constant and  is a function that represents an external force.

Suppose I introduce growth factors that effect my population at a rate of

. At what rate  do I need in order for my population to grow? (Hint: Find  and determine for what  will  increase in time.)

Explanation:

Start by substituting  into the integral to get:

We can combine this into one large term:

Since .

This can only grow when:

Example Question #1021 : Integrals

Evaluate the integral with a substitution,

Explanation:

Let

We can now convert this back to a function of  by substituting

Example Question #677 : Finding Integrals

Calculate the following integral:

Explanation:

Add 2 and subtract 2 from the numerator of the integrand:.

Simplify and apply the difference rule:

Solve the first integral: .

Make the following substitution to solve the second integral:

Apply the substitution to the integral:

Solve the integral:

Combine the answers to the two integrals: .

Solution:

Example Question #2771 : Calculus Ii

Evaluate the Integral:

Explanation:

We use substitution to solve the problem:

Let    and

Therefore:

Example Question #679 : Finding Integrals

Evaluate

Explanation:

Here we use substitution to solve for the integrand.  Let u=sin(x) therefore du= cos(x)dx.  Plug your values back in:

Example Question #680 : Finding Integrals

Explanation:

To integrate this expression, you have to use u substitution. First, assign your u expression:

Now, plug everything back in so you can integrate:

Now integrate:

From here substitute the original variable back into the expression.

Evaluate at 2 and then 1.

Subtract the results:

Example Question #681 : Finding Integrals

Calculate the following integral:

Explanation:

Factor out  from the integrand, and simplify:

Make the following substitution:

Plug the substitution into the integrand:

.

Use the Pythagorean identity to make the following substitution, and simplify:

Apply the following identity to the integrand:

.

Separate the integral into two separate integrals:

.

Solve the first integral:

.

Make the following substitution for the second integral:

.

Apply the substitution, and solve the integral:

.

Solve for

Plug values for   back into solution to integral:

Recall that,

and from above,

Therefore,

.