Evaluate the Integral

Problem

(∫_24^25)(x√(,x2−576)*d(x))

Solution

1. Identify the substitution method to simplify the integrand. Let u=x2−576

2. Differentiate the substitution to find the relationship between d(u) and d(x)

d(u)/d(x)=2*x

d(u)=2*x*d(x)

1/2*d(u)=x*d(x)

3. Change the limits of integration from x to u

When *x=24,u=24−576=576−576=0

When *x=25,u=25−576=625−576=49

4. Substitute the new variables and limits into the integral.

(∫_0^49)(√(,u)⋅1/2*d(u))

1/2*(∫_0^49)(u(1/2)*d(u))

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

1/2*[(u(3/2))/(3/2)]490

1/2⋅2/3*[u(3/2)]490

1/3*[u(3/2)]490

6. Evaluate the definite integral at the boundaries.

1/3*(49(3/2)−0(3/2))

1/3*((√(,49))3−0)

1/3*(7)

343/3

Final Answer

(∫_24^25)(x√(,x2−576)*d(x))=343/3

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