Calculus Examples

Evaluate the Integral integral of 20t^3e^(-t^4) with respect to t
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Let . Then , so . Rewrite using and .
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Step 2.1
Let . Find .
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Step 2.1.1
Differentiate .
Step 2.1.2
Differentiate using the Power Rule which states that is where .
Step 2.2
Rewrite the problem using and .
Step 3
Simplify.
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Step 3.1
Rewrite as .
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Step 3.1.1
Use to rewrite as .
Step 3.1.2
Apply the power rule and multiply exponents, .
Step 3.1.3
Combine and .
Step 3.1.4
Cancel the common factor of and .
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Step 3.1.4.1
Factor out of .
Step 3.1.4.2
Cancel the common factors.
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Step 3.1.4.2.1
Factor out of .
Step 3.1.4.2.2
Cancel the common factor.
Step 3.1.4.2.3
Rewrite the expression.
Step 3.1.4.2.4
Divide by .
Step 3.2
Combine and .
Step 3.3
Combine and .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Simplify.
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Step 5.1
Combine and .
Step 5.2
Cancel the common factor of and .
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Step 5.2.1
Factor out of .
Step 5.2.2
Cancel the common factors.
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Step 5.2.2.1
Factor out of .
Step 5.2.2.2
Cancel the common factor.
Step 5.2.2.3
Rewrite the expression.
Step 5.2.2.4
Divide by .
Step 6
Let . Then , so . Rewrite using and .
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Step 6.1
Let . Find .
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Step 6.1.1
Differentiate .
Step 6.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.1.3
Differentiate using the Power Rule which states that is where .
Step 6.1.4
Multiply by .
Step 6.2
Rewrite the problem using and .
Step 7
Simplify.
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Step 7.1
Move the negative in front of the fraction.
Step 7.2
Combine and .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
Multiply by .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Simplify.
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Step 11.1
Combine and .
Step 11.2
Cancel the common factor of and .
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Step 11.2.1
Factor out of .
Step 11.2.2
Cancel the common factors.
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Step 11.2.2.1
Factor out of .
Step 11.2.2.2
Cancel the common factor.
Step 11.2.2.3
Rewrite the expression.
Step 11.2.2.4
Divide by .
Step 12
The integral of with respect to is .
Step 13
Simplify.
Step 14
Substitute back in for each integration substitution variable.
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Step 14.1
Replace all occurrences of with .
Step 14.2
Replace all occurrences of with .