Evaluate the Integral

Problem

22*(∫_0^1)(x3√(,1−x2)*d(x))

Solution

1. Identify the integral and use a substitution to simplify the integrand. Let 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 x2 term in terms of u using the relation x2=1−u

4. Change the limits of integration. When x=0 u=1−0=1 When x=1 u=1−1=0

5. Substitute these into the integral, including the constant factor of 22.

22*(∫_1^0)((1−u)√(,u)*(−1/2)*d(u))

6. Simplify the expression by using the negative sign to flip the limits of integration and multiplying the constants.

11*(∫_0^1)((1−u)*u(1/2)*d(u))

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

11*(∫_0^1)((u(1/2)−u(3/2))*d(u))

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

11*[2/3*u(3/2)−2/5*u(5/2)]10

9. Evaluate the definite integral at the boundaries 1 and 0.

11*((2/3−2/5)−(0−0))

10. Find a common denominator to subtract the fractions.

11*(10/15−6/15)

11. Multiply the resulting fraction by the constant 11.

11*(4/15)

Final Answer

22*(∫_0^1)(x3√(,1−x2)*d(x))=44/15

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