Calculus Examples

Evaluate the Integral integral from 0 to 1 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
Substitute the lower limit in for in .
Step 5.3
Subtract from .
Step 5.4
Substitute the upper limit in for in .
Step 5.5
Simplify.
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Step 5.5.1
Multiply by .
Step 5.5.2
Subtract from .
Step 5.6
The values found for and will be used to evaluate the definite integral.
Step 5.7
Rewrite the problem using , , and the new limits of integration.
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
Combine and .
Step 8.3
To write as a fraction with a common denominator, multiply by .
Step 8.4
Combine and .
Step 8.5
Combine the numerators over the common denominator.
Step 8.6
Move to the left of .
Step 9
Substitute and simplify.
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Step 9.1
Evaluate at and at .
Step 9.2
Evaluate at and at .
Step 9.3
Simplify.
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Step 9.3.1
Multiply by .
Step 9.3.2
Subtract from .
Step 9.3.3
Rewrite as .
Step 9.3.4
Apply the power rule and multiply exponents, .
Step 9.3.5
Cancel the common factor of .
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Step 9.3.5.1
Cancel the common factor.
Step 9.3.5.2
Rewrite the expression.
Step 9.3.6
Raising to any positive power yields .
Step 9.3.7
Multiply by .
Step 9.3.8
Multiply by .
Step 9.3.9
Cancel the common factor of and .
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Step 9.3.9.1
Factor out of .
Step 9.3.9.2
Cancel the common factors.
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Step 9.3.9.2.1
Factor out of .
Step 9.3.9.2.2
Cancel the common factor.
Step 9.3.9.2.3
Rewrite the expression.
Step 9.3.9.2.4
Divide by .
Step 9.3.10
Multiply by .
Step 9.3.11
Multiply by .
Step 9.3.12
Add and .
Step 9.3.13
One to any power is one.
Step 9.3.14
Multiply by .
Step 9.3.15
Multiply by .
Step 9.3.16
Cancel the common factor of and .
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Step 9.3.16.1
Factor out of .
Step 9.3.16.2
Cancel the common factors.
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Step 9.3.16.2.1
Factor out of .
Step 9.3.16.2.2
Cancel the common factor.
Step 9.3.16.2.3
Rewrite the expression.
Step 9.3.16.2.4
Divide by .
Step 9.3.17
Add and .
Step 9.3.18
Multiply by .
Step 9.3.19
Rewrite as .
Step 9.3.20
Apply the power rule and multiply exponents, .
Step 9.3.21
Cancel the common factor of .
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Step 9.3.21.1
Cancel the common factor.
Step 9.3.21.2
Rewrite the expression.
Step 9.3.22
Raising to any positive power yields .
Step 9.3.23
Multiply by .
Step 9.3.24
Cancel the common factor of and .
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Step 9.3.24.1
Factor out of .
Step 9.3.24.2
Cancel the common factors.
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Step 9.3.24.2.1
Factor out of .
Step 9.3.24.2.2
Cancel the common factor.
Step 9.3.24.2.3
Rewrite the expression.
Step 9.3.24.2.4
Divide by .
Step 9.3.25
One to any power is one.
Step 9.3.26
Multiply by .
Step 9.3.27
Subtract from .
Step 9.3.28
Multiply by .
Step 9.3.29
Combine and .
Step 9.3.30
Multiply by .
Step 9.3.31
Add and .
Step 9.3.32
Rewrite as a product.
Step 9.3.33
Multiply by .
Step 9.3.34
Multiply by .
Step 10
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 11