Evaluate the Integral

Problem

(∫_0^8)(x(1/3)√(,x(4/3)+9)*d(x))

Solution

1. Identify the substitution to simplify the integrand by letting u be the expression inside the square root.

u=x(4/3)+9

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

d(u)/d(x)=4/3*x(1/3)

d(u)=4/3*x(1/3)*d(x)

3/4*d(u)=x(1/3)*d(x)

3. Change the limits of integration from x to u using the substitution formula.

Lower limit: *x=0⇒u=0(4/3)+9=9

Upper limit: *x=8⇒u=8(4/3)+9=(2)(4/3)+9=16+9=25

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

(∫_9^25)(√(,u)⋅3/4*d(u))

3/4*(∫_9^25)(u(1/2)*d(u))

5. Integrate the expression using the power rule for integration.

3/4*[(u(3/2))/(3/2)]259

3/4⋅2/3*[u(3/2)]259

1/2*[u(3/2)]259

6. Evaluate the definite integral by plugging in the upper and lower limits.

1/2*(25(3/2)−9(3/2))

1/2*(125−27)

1/2*(98)

49

Final Answer

(∫_0^8)(x(1/3)√(,x(4/3)+9)*d(x))=49

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