Calculus Examples

Evaluate the Integral integral from 1 to 2 of (e^(1/(x^4)))/(x^5) 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
Apply basic rules of exponents.
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Step 1.1.2.1
Rewrite as .
Step 1.1.2.2
Multiply the exponents in .
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Step 1.1.2.2.1
Apply the power rule and multiply exponents, .
Step 1.1.2.2.2
Multiply by .
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
Rewrite the expression using the negative exponent rule .
Step 1.1.4.2
Combine terms.
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Step 1.1.4.2.1
Combine and .
Step 1.1.4.2.2
Move the negative in front of the fraction.
Step 1.2
Substitute the lower limit in for in .
Step 1.3
Simplify.
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Step 1.3.1
One to any power is one.
Step 1.3.2
Divide by .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
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
Since is constant with respect to , move out of the integral.
Step 3
The integral of with respect to is .
Step 4
Substitute and simplify.
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Step 4.1
Evaluate at and at .
Step 4.2
Simplify.
Step 5
Simplify.
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Step 5.1
Apply the distributive property.
Step 5.2
Combine and .
Step 5.3
Multiply .
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Step 5.3.1
Multiply by .
Step 5.3.2
Multiply by .
Step 5.3.3
Combine and .
Step 6
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 7