Evaluate the Integral

Problem

(∫_^)(x√(,1−x)*d(x))

Solution

1. Identify the substitution method to simplify the integrand. Let u=1−x

2. Differentiate the substitution to find the relationship between d(u) and d(x) Since u=1−x then d(u)=−d(x) which means d(x)=−d(u)

3. Solve for x in terms of u From u=1−x we get x=1−u

4. Substitute the expressions for x √(,1−x) and d(x) into the integral.

(∫_^)((1−u)√(,u)*(−d(u)))

5. Distribute the negative sign and the √(,u) (which is u(1/2) into the parentheses.

(∫_^)((u−1)*u(1/2)*d(u))

(∫_^)((u(3/2)−u(1/2))*d(u))

6. Integrate each term using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

(u(5/2))/(5/2)−(u(3/2))/(3/2)+C

7. Simplify the coefficients.

2/5*u(5/2)−2/3*u(3/2)+C

8. Back-substitute u=1−x to return to the original variable.

2/5*(1−x)(5/2)−2/3*(1−x)(3/2)+C

Final Answer

(∫_^)(x√(,1−x)*d(x))=2/5*(1−x)(5/2)−2/3*(1−x)(3/2)+C

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