Calculus Examples

Evaluate the Integral integral from 1 to e of ( natural log of x)/x with respect to x
Step 1
Let . Then , so . Rewrite using and .
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Step 1.1
Let . Find .
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Step 1.1.1
Differentiate .
Step 1.1.2
The derivative of with respect to is .
Step 1.2
Substitute the lower limit in for in .
Step 1.3
The natural logarithm of is .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
The natural logarithm of is .
Step 1.6
The values found for and will be used to evaluate the definite integral.
Step 1.7
Rewrite the problem using , , and the new limits of integration.
Step 2
By the Power Rule, the integral of with respect to is .
Step 3
Substitute and simplify.
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Step 3.1
Evaluate at and at .
Step 3.2
Simplify.
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Step 3.2.1
One to any power is one.
Step 3.2.2
Multiply by .
Step 3.2.3
Raising to any positive power yields .
Step 3.2.4
Multiply by .
Step 3.2.5
Multiply by .
Step 3.2.6
Add and .
Step 4
The result can be shown in multiple forms.
Exact Form:
Decimal Form: