Calculus Examples

Evaluate the Integral integral from -2 to -1 of 2x(6-x^2)^3 with respect to x
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.
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Step 2.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3
Evaluate .
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Step 2.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.2
Differentiate using the Power Rule which states that is where .
Step 2.1.3.3
Multiply by .
Step 2.1.4
Subtract from .
Step 2.2
Substitute the lower limit in for in .
Step 2.3
Simplify.
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Step 2.3.1
Simplify each term.
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Step 2.3.1.1
Raise to the power of .
Step 2.3.1.2
Multiply by .
Step 2.3.2
Subtract from .
Step 2.4
Substitute the upper limit in for in .
Step 2.5
Simplify.
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Step 2.5.1
Simplify each term.
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Step 2.5.1.1
Multiply by by adding the exponents.
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Step 2.5.1.1.1
Multiply by .
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Step 2.5.1.1.1.1
Raise to the power of .
Step 2.5.1.1.1.2
Use the power rule to combine exponents.
Step 2.5.1.1.2
Add and .
Step 2.5.1.2
Raise to the power of .
Step 2.5.2
Subtract from .
Step 2.6
The values found for and will be used to evaluate the definite integral.
Step 2.7
Rewrite the problem using , , and the new limits of integration.
Step 3
Simplify.
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Step 3.1
Move the negative in front of the fraction.
Step 3.2
Combine and .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Multiply by .
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Simplify.
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Step 7.1
Combine and .
Step 7.2
Cancel the common factor of and .
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Step 7.2.1
Factor out of .
Step 7.2.2
Cancel the common factors.
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Step 7.2.2.1
Factor out of .
Step 7.2.2.2
Cancel the common factor.
Step 7.2.2.3
Rewrite the expression.
Step 7.2.2.4
Divide by .
Step 8
By the Power Rule, the integral of with respect to is .
Step 9
Combine and .
Step 10
Substitute and simplify.
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Step 10.1
Evaluate at and at .
Step 10.2
Simplify.
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Step 10.2.1
Raise to the power of .
Step 10.2.2
Raise to the power of .
Step 10.2.3
Cancel the common factor of and .
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Step 10.2.3.1
Factor out of .
Step 10.2.3.2
Cancel the common factors.
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Step 10.2.3.2.1
Factor out of .
Step 10.2.3.2.2
Cancel the common factor.
Step 10.2.3.2.3
Rewrite the expression.
Step 10.2.3.2.4
Divide by .
Step 10.2.4
Multiply by .
Step 10.2.5
To write as a fraction with a common denominator, multiply by .
Step 10.2.6
Combine and .
Step 10.2.7
Combine the numerators over the common denominator.
Step 10.2.8
Simplify the numerator.
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Step 10.2.8.1
Multiply by .
Step 10.2.8.2
Subtract from .
Step 11
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Mixed Number Form:
Step 12