Calculus Examples

Evaluate the Integral integral from 0 to 1/6 of 3/( square root of 1-9x^2) with respect to x
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Write the expression using exponents.
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Step 2.1
Rewrite as .
Step 2.2
Rewrite as .
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
Since is constant with respect to , the derivative of with respect to is .
Step 3.1.3
Differentiate using the Power Rule which states that is where .
Step 3.1.4
Multiply by .
Step 3.2
Substitute the lower limit in for in .
Step 3.3
Multiply by .
Step 3.4
Substitute the upper limit in for in .
Step 3.5
Cancel the common factor of .
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Step 3.5.1
Factor out of .
Step 3.5.2
Cancel the common factor.
Step 3.5.3
Rewrite the expression.
Step 3.6
The values found for and will be used to evaluate the definite integral.
Step 3.7
Rewrite the problem using , , and the new limits of integration.
Step 4
Since is constant with respect to , move out of the integral.
Step 5
The integral of with respect to is
Step 6
Simplify.
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Step 6.1
Combine and .
Step 6.2
Combine and .
Step 6.3
Cancel the common factor of .
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Step 6.3.1
Cancel the common factor.
Step 6.3.2
Divide by .
Step 7
Evaluate at and at .
Step 8
Simplify.
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Step 8.1
Simplify each term.
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Step 8.1.1
The exact value of is .
Step 8.1.2
The exact value of is .
Step 8.1.3
Multiply by .
Step 8.2
Add and .
Step 9
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 10