Calculus Examples

Evaluate the Integral integral of x(x-4)^9 with respect to x
Step 1
Let . Then . 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
By the Sum Rule, the derivative of with respect to is .
Step 1.1.3
Differentiate using the Power Rule which states that is where .
Step 1.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.5
Add and .
Step 1.2
Rewrite the problem using and .
Step 2
Multiply .
Step 3
Multiply by by adding the exponents.
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Step 3.1
Multiply by .
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Step 3.1.1
Raise to the power of .
Step 3.1.2
Use the power rule to combine exponents.
Step 3.2
Add and .
Step 4
Split the single integral into multiple integrals.
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Since is constant with respect to , move out of the integral.
Step 7
By the Power Rule, the integral of with respect to is .
Step 8
Simplify.
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Step 8.1
Simplify.
Step 8.2
Simplify.
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Step 8.2.1
Combine and .
Step 8.2.2
Cancel the common factor of and .
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Step 8.2.2.1
Factor out of .
Step 8.2.2.2
Cancel the common factors.
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Step 8.2.2.2.1
Factor out of .
Step 8.2.2.2.2
Cancel the common factor.
Step 8.2.2.2.3
Rewrite the expression.
Step 9
Replace all occurrences of with .