Evaluate the Integral

Problem

8*(∫_0^1)(x3√(,1−x2)*d(x))

Solution

1. Apply substitution to simplify the integrand by letting u=1−x2

2. Calculate the differential d(u)=−2*x*d(x) which implies x*d(x)=−1/2*d(u)

3. Express the remaining terms in terms of u using x2=1−u

4. Change the limits of integration from x to u when x=0 u=1 when x=1 u=0

5. Substitute into the integral to rewrite the expression:

8*(∫_1^0)((1−u)√(,u)*(−1/2)*d(u))

6. Simplify the constant and reverse the limits of integration to remove the negative sign:

4*(∫_0^1)((1−u)*u(1/2)*d(u))

7. Distribute the term u(1/2) inside the parentheses:

4*(∫_0^1)((u(1/2)−u(3/2))*d(u))

8. Integrate term by term using the power rule:

4*[2/3*u(3/2)−2/5*u(5/2)]10

9. Evaluate at the boundaries 1 and 0

4*((2/3−2/5)−(0−0))

10. Find a common denominator and simplify the final value:

4*(10/15−6/15)=4*(4/15)=16/15

Final Answer

8*(∫_0^1)(x3√(,1−x2)*d(x))=16/15

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