# Evaluate integral of square root of 25-x^2 with respect to x

Let , where . Then . Note that since , is positive.
Simplify terms.
Simplify .
Simplify each term.
Apply the product rule to .
Raise to the power of .
Multiply by .
Factor out of .
Factor out of .
Factor out of .
Apply pythagorean identity.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Simplify.
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Since is constant with respect to , move out of the integral.
Use the halfangle formula to rewrite as .
Since is constant with respect to , move out of the integral.
Combine and .
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
Let . Then , so . Rewrite using and .
Let . Find .
Rewrite.
Divide by .
Rewrite the problem using and .
Combine and .
Since is constant with respect to , move out of the integral.
The integral of with respect to is .
Simplify.
Substitute back in for each integration substitution variable.
Replace all occurrences of with .
Replace all occurrences of with .
Replace all occurrences of with .
Combine and .
Apply the distributive property.
Combine and .
Multiply .
Multiply and .
Multiply by .
Reorder terms.
Evaluate integral of square root of 25-x^2 with respect to x