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NARRATOR: Exercise 13.

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Differentiate f of x equals
x times log e of x minus x,

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and hence find the indefinite
integral of log e of x.

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So the derivative of
x times log e of x minus x

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is going to require the product rule

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to differentiate the first part
of this expression.

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So the derivative of x is 1
and multiply it by log e of x

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plus x times the derivative
of log e of x, which is one on x,

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and then minus the derivative of x,
which is 1.

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Simplifying each part gives
log e of x plus 1 minus 1,

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which clearly simplifies
to log e of x.

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So we see that the derivative of x
times log e of x minus x

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is log e of x.

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And if the derivative of x times
log e of x minus x is log e of x,

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then we can reverse that statement
and say that

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the antiderivative, or the
indefinite integral, of log e of x

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is x times log e of x minus x.

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And, of course, because
it's an indefinite integral,

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we need to add the plus c.