Calculus Examples

Evaluate the Integral integral from 0 to natural log of 2 of e^(3x) 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
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3
Differentiate using the Power Rule which states that is where .
Step 1.1.4
Multiply by .
Step 1.2
Substitute the lower limit in for in .
Step 1.3
Multiply by .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Simplify.
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Step 1.5.1
Simplify by moving inside the logarithm.
Step 1.5.2
Raise to the power of .
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
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
The integral of with respect to is .
Step 5
Substitute and simplify.
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Step 5.1
Evaluate at and at .
Step 5.2
Simplify.
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Step 5.2.1
Exponentiation and log are inverse functions.
Step 5.2.2
Anything raised to is .
Step 5.2.3
Multiply by .
Step 5.2.4
Subtract from .
Step 5.2.5
Combine and .
Step 6
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Mixed Number Form:
Step 7