Calculus Examples

Evaluate the Integral integral from 0 to 1 of xe^(x^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 using the Power Rule which states that is where .
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
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
Simplify.
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
Apply the constant rule.
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
Combine and .
Step 3.2.2
Multiply by .
Step 4
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 5