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

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Stumbling Robot

A Fraction of a Dot
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Tag: e

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Evaluate the indefinite integral of *x*^{3}e^{-x2}

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Evaluate the indefinite integral of * e*^{x1/2}

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Evaluate the indefinite integral of *x*^{2}e^{-2x}

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Evaluate the indefinite integral of *x*^{2}e^{x}

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Evaluate the indefinite integral of *xe*^{-x}

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Evaluate the indefinite integral of * xe*^{x}

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Find the derivative of *e*^{eex}

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Find the derivative of * e*^{(ex)}

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Find the derivative of *e*^{log x}

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Find the derivative of *e*^{cos2x}

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

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

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

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,

Compute the following indefinite integral,

To evaluate this integral we use integration by parts, letting

Therefore,

Compute the following indefinite integral:

To evaluate this integral we use integration by parts, letting

Then we have,

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 the following function:

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

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 the following function:

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