Evaluate the Integral

Problem

(∫_1^7)(x√(,4*x2+9)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand. Let u=4*x2+9

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

d(u)/d(x)=8*x

d(u)=8*x*d(x)

1/8*d(u)=x*d(x)

3. Determine the new limits of integration by substituting the original x values into the equation for u

Lower limit: *u=4*(1)2+9=13

Upper limit: *u=4*(7)2+9=4*(49)+9=205

4. Substitute the variables and limits into the integral.

(∫_13^205)(1/8√(,u)*d(u))

5. Apply the power rule for integration, where (∫_^)(un*d(u))=(u(n+1))/(n+1)

1/8*(∫_13^205)(u(1/2)*d(u))

1/8*[2/3*u(3/2)]20513

1/12*[u(3/2)]20513

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

1/12*(205(3/2)−13(3/2))

1/12*(205√(,205)−13√(,13))

Final Answer

(∫_1^7)(x√(,4*x2+9)*d(x))=(205√(,205)−13√(,13))/12

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