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Programmed tutorial: Integration of an exponential with base 5: Example 3

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We wish to find the integral Integral of a base 5 exponential of a squae root factor. . We need to be careful to select the correct form for this problem. Is it a Power Rule problem?

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No, x would be in the base and the exponent would be a constant for that form:

is the form for the Power Rule.

Can we use the exponential form with base e?

No, that applies only when the base is e:

How about the exponential form with a general constant base:

Yes, that is what we need to use.

Write the integral of our general form:

Note that this reduces to the correct result when a is equal to e, since ln(e) = 1.

What is the value of a in this problem?

a = 5

What is the value of u in this case?

We have

What is the value of du?

Since we have only

in the original problem, we need to solve for dx. Do that.

Use these reults to set up the integral for this case in terms of u.

Can we take the 2/3 factor outside the integral?

Yes.

Do that.

Write the result in terms of u.

Replace u.

Finally, we have

Let’s check our result by differentiating.

Initially, we get

Complete the differentiation and cancel.

We get

, which checks.

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