Calculus Examples

Evaluate the Integral integral of x^3 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
Factor out .
Step 4
Using the Pythagorean Identity, rewrite as .
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
The derivative of with respect to is .
Step 5.2
Rewrite the problem using and .
Step 6
Multiply .
Step 7
Simplify.
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Step 7.1
Rewrite as .
Step 7.2
Multiply by by adding the exponents.
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Step 7.2.1
Use the power rule to combine exponents.
Step 7.2.2
Add and .
Step 8
Split the single integral into multiple integrals.
Step 9
Since is constant with respect to , move out of the integral.
Step 10
By the Power Rule, the integral of with respect to is .
Step 11
By the Power Rule, the integral of with respect to is .
Step 12
Simplify.
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Step 12.1
Combine and .
Step 12.2
Simplify.
Step 13
Substitute back in for each integration substitution variable.
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Step 13.1
Replace all occurrences of with .
Step 13.2
Replace all occurrences of with .