Home » e » Page 3

# Evaluate the indefinite integral of x3e-x2

Compute the following indefinite integral,

First, we rearrange the integrand a bit to get a form in which we can make a -substitution,

Then, let

Therefore, making a -substitution, we have

# Evaluate the indefinite integral of ex1/2

Compute the following indefinite integral

To evaluate this integral we want to make a substitution. First multiply the numerator and denominator by to obtain,

Now define the function by

This implies

Therefore, using the method of substitution, we have

# Evaluate the indefinite integral of x2e-2x

Compute the following indefinite integral,

To compute this integral we will integrate by parts twice. First, let

Therefore we have

To evaluate this next integral we use integration by parts a second time with

Giving us

So, putting this back into our formula we have

# Evaluate the indefinite integral of x2ex

Compute the following indefinite integral,

We compute the integral using integration by parts. Let,

Then we have

But we know from a previous exercise (Section 6.17, Exercise #13) that

Therefore we have,

# Evaluate the indefinite integral of xe-x

Compute the following indefinite integral,

To evaluate this integral we use integration by parts, letting

Therefore,

# Evaluate the indefinite integral of xex

Compute the following indefinite integral:

To evaluate this integral we use integration by parts, letting

Then we have,

# Find the derivative of eeex

Find the derivative of the following function:

Using the chain rule and the formula for the derivative of an exponential we compute,

From the previous exercise (Section 6.17, Exercise #11) (or by applying the chain rule and the formula for the derivative of the exponential again) we know

Therefore,

# Find the derivative of e(ex)

Find the derivative of the following function:

Again, we use the chain rule and the formula for the derivative of the exponential,

# Find the derivative of elog x

Find the derivative of the following function:

Since

the derivative is given by

We could check this if we wanted by using the chain rule and the derivative of the exponential on the original formula for ,

# Find the derivative of ecos2x

Find the derivative of the following function:

Using the chain rule and the formula for the derivative of the exponential we have,