Evaluate the Integral

Problem

(∫_^)(x√(,1−x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function 1−x2 is a multiple of the outer factor x

2. Substitute u=1−x2 to simplify the integrand.

3. Differentiate u with respect to x to find d(u)=−2*x*d(x) which implies x*d(x)=−1/2*d(u)

4. Rewrite the integral in terms of u by substituting the expressions found in the previous steps.

(∫_^)(√(,u)*(−1/2)*d(u))

5. Factor out the constant −1/2 from the integral.

−1/2*(∫_^)(u(1/2)*d(u))

6. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

−1/2*((u(3/2))/(3/2))+C

7. Simplify the resulting expression by multiplying the fractions.

−1/3*u(3/2)+C

8. Back-substitute the original expression 1−x2 for u to get the final result in terms of x

Final Answer

(∫_^)(x√(,1−x2)*d(x))=−1/3*(1−x2)(3/2)+C

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