Evaluate the Integral

Problem

(∫_^)(√(,1−x)*d(x))

Solution

1. Identify the integral as a composition of functions, suggesting the use of usubstitution.

2. Substitute u=1−x to simplify the radicand.

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

4. Rewrite the integral in terms of u by substituting the expressions for 1−x and d(x)

(∫_^)(√(,u)*(−d(u)))

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

−(∫_^)(u(1/2)*d(u))

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

−(u(3/2))/(3/2)+C

7. Simplify the coefficient by multiplying by the reciprocal of the exponent's denominator.

−2/3*u(3/2)+C

8. Back-substitute the original expression 1−x for u to obtain the final result.

Final Answer

(∫_^)(√(,1−x)*d(x))=−2/3*(1−x)(3/2)+C

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