Evaluate the Integral

Problem

(∫_0^√(,11))((10*x)/√(,x2+25)*d(x))

Solution

1. Identify the substitution method to simplify the integrand. Let u=x2+25

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)

3. Rewrite the numerator 10*x*d(x) in terms of d(u)

10*x*d(x)=5*(2*x*d(x))=5*d(u)

4. Determine the new limits of integration for u When x=0 u=0+25=25 When x=√(,11) u=(√(,11))2+25=11+25=36

5. Substitute the variables and limits into the integral.

(∫_25^36)(5/√(,u)*d(u))

6. Rewrite the integrand using a fractional exponent for easier integration.

(∫_25^36)(5*u(−1/2)*d(u))

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

[5⋅(u(1/2))/(1/2)]3625

[10√(,u)]3625

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

10√(,36)−10√(,25)

10*(6)−10*(5)

60−50=10

Final Answer

(∫_0^√(,11))((10*x)/√(,x2+25)*d(x))=10

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