Evaluate the Integral

Problem

(∫_0^5)(x√(,25−x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function 25−x2 is proportional to the outer factor x

2. Define the substitution variable u=25−x2

3. Calculate the differential d(u) by differentiating u with respect to x

d(u)/d(x)=−2*x

d(u)=−2*x*d(x)

−1/2*d(u)=x*d(x)

4. Determine the new limits of integration for u

When *x=0,u=25−0=25

When *x=5,u=25−5=0

5. Substitute the variables and limits into the integral.

(∫_25^0)(−1/2√(,u)*d(u))

6. Simplify the integral by using the property (∫_a^b)(ƒ(u)*d(u))=−(∫_b^a)(ƒ(u)*d(u)) to flip the limits and remove the negative sign.

1/2*(∫_0^25)(u(1/2)*d(u))

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

1/2*[(u(3/2))/(3/2)]250

1/2⋅2/3*[u(3/2)]250

1/3*[u(3/2)]250

8. Evaluate the expression at the upper and lower limits.

1/3*(25(3/2)−0(3/2))

1/3*((√(,25))3−0)

1/3*(5)

125/3

Final Answer

(∫_0^5)(x√(,25−x2)*d(x))=125/3

Want more problems? Check here!