Calculus Examples

Evaluate the Integral integral from 2 to 3 of 1+2e^(-0.4x) with respect to x
Step 1
Split the single integral into multiple integrals.
Step 2
Apply the constant rule.
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Let . Then , so . Rewrite using and .
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Step 4.1
Let . Find .
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Step 4.1.1
Differentiate .
Step 4.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3
Differentiate using the Power Rule which states that is where .
Step 4.1.4
Multiply by .
Step 4.2
Substitute the lower limit in for in .
Step 4.3
Multiply by .
Step 4.4
Substitute the upper limit in for in .
Step 4.5
Multiply by .
Step 4.6
The values found for and will be used to evaluate the definite integral.
Step 4.7
Rewrite the problem using , , and the new limits of integration.
Step 5
Simplify.
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Step 5.1
Move the negative in front of the fraction.
Step 5.2
Combine and .
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Multiply by .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
Simplify.
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Step 9.1
Combine and .
Step 9.2
Move the negative in front of the fraction.
Step 10
The integral of with respect to is .
Step 11
Substitute and simplify.
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Step 11.1
Evaluate at and at .
Step 11.2
Evaluate at and at .
Step 11.3
Subtract from .
Step 12
Simplify.
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Step 12.1
Simplify each term.
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Step 12.1.1
Divide by .
Step 12.1.2
Multiply by .
Step 12.1.3
Apply the distributive property.
Step 12.1.4
Multiply by .
Step 12.2
Simplify each term.
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Step 12.2.1
Rewrite the expression using the negative exponent rule .
Step 12.2.2
Combine and .
Step 12.2.3
Move the negative in front of the fraction.
Step 12.2.4
Rewrite the expression using the negative exponent rule .
Step 12.2.5
Combine and .
Step 13
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 14