Calculus Examples

Evaluate the Integral integral of x square root of 1-x with respect to x
Step 1
Integrate by parts using the formula , where and .
Step 2
Simplify.
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Step 2.1
Combine and .
Step 2.2
Combine and .
Step 2.3
Move to the left of .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Simplify.
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Step 4.1
Multiply by .
Step 4.2
Multiply by .
Step 5
Let . Then , so . Rewrite using and .
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Step 5.1
Let . Find .
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Step 5.1.1
Differentiate .
Step 5.1.2
Differentiate.
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Step 5.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 5.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3
Evaluate .
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Step 5.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3.2
Differentiate using the Power Rule which states that is where .
Step 5.1.3.3
Multiply by .
Step 5.1.4
Subtract from .
Step 5.2
Rewrite the problem using and .
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
Combine and .
Step 8.2
Rewrite as .
Step 8.3
Simplify.
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Step 8.3.1
Combine and .
Step 8.3.2
Combine and .
Step 8.3.3
Move to the left of .
Step 8.3.4
Move to the left of .
Step 8.3.5
Combine and .
Step 8.3.6
Multiply by .
Step 8.3.7
Multiply by .
Step 8.3.8
Multiply by .
Step 8.3.9
To write as a fraction with a common denominator, multiply by .
Step 8.3.10
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 8.3.10.1
Multiply by .
Step 8.3.10.2
Multiply by .
Step 8.3.11
Combine the numerators over the common denominator.
Step 8.3.12
Multiply by .
Step 9
Replace all occurrences of with .
Step 10
Reorder terms.