Evaluate ∫ ln (8x) dx.

Step-by-step explanation:

So we have the integral:

First, let's do a substitution to simplify the inside of natural log. Let y equal to 8x. So:

Differentiate with respect to x:

Divide both sides by 8:

Substitute this into our original integral. Therefore:

Move the co-efficient outside:

Now, perform integration by parts. Integration by parts is as follows:

Note that our integral is the same as saying:

Let u equal the natural log and let dv equal 1 dy. Therefore:

Differentiate:

And let dv equal 1 dy. Thus:

Integrate:

Perform integration by parts:

Simplify:

Evaluate the integral:

Substitute 8x for y:

Distribute:

Constant of Integration:

And we have our answer.

Edit: Typo

9 sandwiches per hour

step-by-step explanation:

when we have a rate, we want a 1 in the denominator

27/3 = 9

27 sandwiches/ 3 hours = 9 sandwiches/hour

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step-by-step explanation: