Calculus Examples

Evaluate the Integral integral from 0 to 1 of xe^(-9x^2) 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
Differentiate using the chain rule, which states that is where and .
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Step 1.1.2.1
To apply the Chain Rule, set as .
Step 1.1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.2.3
Replace all occurrences of with .
Step 1.1.3
Differentiate.
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Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Multiply by .
Step 1.1.4
Simplify.
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Step 1.1.4.1
Reorder the factors of .
Step 1.1.4.2
Reorder factors in .
Step 1.2
Substitute the lower limit in for in .
Step 1.3
Simplify.
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Step 1.3.1
Raising to any positive power yields .
Step 1.3.2
Multiply by .
Step 1.3.3
Anything raised to is .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Simplify.
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Step 1.5.1
One to any power is one.
Step 1.5.2
Multiply by .
Step 1.5.3
Rewrite the expression using the negative exponent rule .
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
Move the negative in front of the fraction.
Step 3
Apply the constant rule.
Step 4
Substitute and simplify.
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Step 4.1
Evaluate at and at .
Step 4.2
Simplify.
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Step 4.2.1
Multiply by .
Step 4.2.2
Move to the left of .
Step 4.2.3
Multiply by .
Step 5
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 6