Calculus Examples

Find the Antiderivative f(x)=x(2-x)^2
Step 1
The function can be found by finding the indefinite integral of the derivative .
Step 2
Set up the integral to solve.
Step 3
Let . Then , so . Rewrite using and .
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Step 3.1
Let . Find .
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Step 3.1.1
Differentiate .
Step 3.1.2
Differentiate.
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Step 3.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 3.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.1.3
Evaluate .
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Step 3.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.1.3.2
Differentiate using the Power Rule which states that is where .
Step 3.1.3.3
Multiply by .
Step 3.1.4
Subtract from .
Step 3.2
Rewrite the problem using and .
Step 4
Expand .
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Step 4.1
Apply the distributive property.
Step 4.2
Apply the distributive property.
Step 4.3
Move parentheses.
Step 4.4
Multiply by .
Step 4.5
Multiply by .
Step 4.6
Raise to the power of .
Step 4.7
Use the power rule to combine exponents.
Step 4.8
Add and .
Step 4.9
Multiply by .
Step 5
Split the single integral into multiple integrals.
Step 6
By the Power Rule, the integral of with respect to is .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
By the Power Rule, the integral of with respect to is .
Step 9
Simplify.
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Step 9.1
Simplify.
Step 9.2
Simplify.
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Step 9.2.1
Combine and .
Step 9.2.2
Move the negative in front of the fraction.
Step 10
Replace all occurrences of with .
Step 11
The answer is the antiderivative of the function .