Calculus Examples

Evaluate the Integral integral of x^2 square root of 1-x^2 with respect to x
Step 1
Let , where . Then . Note that since , is positive.
Step 2
Simplify terms.
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Step 2.1
Simplify .
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Step 2.1.1
Apply pythagorean identity.
Step 2.1.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2
Simplify.
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Step 2.2.1
Raise to the power of .
Step 2.2.2
Raise to the power of .
Step 2.2.3
Use the power rule to combine exponents.
Step 2.2.4
Add and .
Step 3
Use the half-angle formula to rewrite as .
Step 4
Use the half-angle formula to rewrite as .
Step 5
Simplify.
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Step 5.1
Multiply by .
Step 5.2
Multiply by .
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Let . Then , so . Rewrite using and .
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Step 7.1
Let . Find .
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Step 7.1.1
Differentiate .
Step 7.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 7.1.3
Differentiate using the Power Rule which states that is where .
Step 7.1.4
Multiply by .
Step 7.2
Rewrite the problem using and .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
Simplify by multiplying through.
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Step 9.1
Simplify.
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Step 9.1.1
Multiply by .
Step 9.1.2
Multiply by .
Step 9.2
Expand .
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Step 9.2.1
Apply the distributive property.
Step 9.2.2
Apply the distributive property.
Step 9.2.3
Apply the distributive property.
Step 9.2.4
Move .
Step 9.2.5
Multiply by .
Step 9.2.6
Multiply by .
Step 9.2.7
Multiply by .
Step 9.2.8
Factor out negative.
Step 9.2.9
Raise to the power of .
Step 9.2.10
Raise to the power of .
Step 9.2.11
Use the power rule to combine exponents.
Step 9.2.12
Add and .
Step 9.2.13
Subtract from .
Step 9.2.14
Subtract from .
Step 10
Split the single integral into multiple integrals.
Step 11
Apply the constant rule.
Step 12
Since is constant with respect to , move out of the integral.
Step 13
Use the half-angle formula to rewrite as .
Step 14
Since is constant with respect to , move out of the integral.
Step 15
Split the single integral into multiple integrals.
Step 16
Apply the constant rule.
Step 17
Let . Then , so . Rewrite using and .
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Step 17.1
Let . Find .
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Step 17.1.1
Differentiate .
Step 17.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 17.1.3
Differentiate using the Power Rule which states that is where .
Step 17.1.4
Multiply by .
Step 17.2
Rewrite the problem using and .
Step 18
Combine and .
Step 19
Since is constant with respect to , move out of the integral.
Step 20
The integral of with respect to is .
Step 21
Simplify.
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Step 21.1
Simplify.
Step 21.2
Simplify.
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Step 21.2.1
To write as a fraction with a common denominator, multiply by .
Step 21.2.2
Combine and .
Step 21.2.3
Combine the numerators over the common denominator.
Step 21.2.4
Move to the left of .
Step 21.2.5
Subtract from .
Step 22
Substitute back in for each integration substitution variable.
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Step 22.1
Replace all occurrences of with .
Step 22.2
Replace all occurrences of with .
Step 22.3
Replace all occurrences of with .
Step 22.4
Replace all occurrences of with .
Step 22.5
Replace all occurrences of with .
Step 23
Simplify.
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Step 23.1
Simplify each term.
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Step 23.1.1
Cancel the common factor of .
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Step 23.1.1.1
Cancel the common factor.
Step 23.1.1.2
Divide by .
Step 23.1.2
Multiply by .
Step 23.2
Apply the distributive property.
Step 23.3
Combine and .
Step 23.4
Multiply .
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Step 23.4.1
Multiply by .
Step 23.4.2
Multiply by .
Step 24
Reorder terms.