Evaluate the Integral

Problem

(∫_1^6)(x√(,5*x2+4)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function 5*x2+4 is a multiple of the outer factor x

2. Define the substitution variable u=5*x2+4

3. Differentiate u with respect to x to find d(u)=10*x*d(x) which implies x*d(x)=1/10*d(u)

4. Change the limits of integration from x to u When x=1 u=5*(1)2+4=9 When x=6 u=5*(6)2+4=184

5. Substitute the variables and limits into the integral.

(∫_9^184)(1/10√(,u)*d(u))

6. Apply the power rule for integration, where (∫_^)(u(1/2)*d(u))=2/3*u(3/2)

1/10*[2/3*u(3/2)]1849

7. Simplify the constant coefficient.

1/15*[u(3/2)]1849

8. Evaluate at the upper and lower limits.

1/15*(184(3/2)−9(3/2))

9. Calculate the numerical values, noting 9(3/2)=27 and 184(3/2)=184√(,184)=184√(,4⋅46)=368√(,46)

(368√(,46)−27)/15

Final Answer

(∫_1^6)(x√(,5*x2+4)*d(x))=(368√(,46)−27)/15

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