Calculus Examples

Evaluate the Integral integral of (x^2)/( square root of 16-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
Simplify each term.
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Step 2.1.1.1
Apply the product rule to .
Step 2.1.1.2
Raise to the power of .
Step 2.1.1.3
Multiply by .
Step 2.1.2
Factor out of .
Step 2.1.3
Factor out of .
Step 2.1.4
Factor out of .
Step 2.1.5
Apply pythagorean identity.
Step 2.1.6
Rewrite as .
Step 2.1.7
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2
Reduce the expression by cancelling the common factors.
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Step 2.2.1
Cancel the common factor of .
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Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Rewrite the expression.
Step 2.2.2
Simplify.
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Step 2.2.2.1
Factor out of .
Step 2.2.2.2
Apply the product rule to .
Step 2.2.2.3
Raise to the power of .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Use the half-angle formula to rewrite as .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Simplify.
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Step 6.1
Combine and .
Step 6.2
Cancel the common factor of and .
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Step 6.2.1
Factor out of .
Step 6.2.2
Cancel the common factors.
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Step 6.2.2.1
Factor out of .
Step 6.2.2.2
Cancel the common factor.
Step 6.2.2.3
Rewrite the expression.
Step 6.2.2.4
Divide by .
Step 7
Split the single integral into multiple integrals.
Step 8
Apply the constant rule.
Step 9
Since is constant with respect to , move out of the integral.
Step 10
Let . Then , so . Rewrite using and .
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Step 10.1
Let . Find .
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Step 10.1.1
Differentiate .
Step 10.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 10.1.3
Differentiate using the Power Rule which states that is where .
Step 10.1.4
Multiply by .
Step 10.2
Rewrite the problem using and .
Step 11
Combine and .
Step 12
Since is constant with respect to , move out of the integral.
Step 13
The integral of with respect to is .
Step 14
Simplify.
Step 15
Substitute back in for each integration substitution variable.
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Step 15.1
Replace all occurrences of with .
Step 15.2
Replace all occurrences of with .
Step 15.3
Replace all occurrences of with .
Step 16
Simplify.
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Step 16.1
Combine and .
Step 16.2
Apply the distributive property.
Step 16.3
Cancel the common factor of .
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Step 16.3.1
Move the leading negative in into the numerator.
Step 16.3.2
Factor out of .
Step 16.3.3
Cancel the common factor.
Step 16.3.4
Rewrite the expression.
Step 16.4
Multiply by .
Step 17
Reorder terms.