Evaluate the Integral

Problem

(∫_0^5)(√(,x2+25)*d(x))

Solution

1. Identify the integral form as √(,x2+a2) where a=5 which suggests a trigonometric substitution.

2. Substitute x=5*tan(θ) which implies d(x)=5*sec2(θ)*d(θ)

3. Determine the new limits of integration: when x=0 θ=0 when x=5 θ=π/4

4. Simplify the integrand using the identity √(,25*tan2(θ)+25)=5*sec(θ)

5. Rewrite the integral in terms of θ

(∫_0^π/4)((5*sec(θ))*(5*sec2(θ))*d(θ))=25*(∫_0^π/4)(sec3(θ)*d(θ))

6. Apply the reduction formula for the integral of sec3(θ)

(∫_^)(sec3(θ)*d(θ))=1/2*(sec(θ)*tan(θ)+ln(sec(θ)+tan(θ)))

7. Evaluate the definite integral from 0 to π/4

25*[1/2*(sec(θ)*tan(θ)+ln(sec(θ)+tan(θ)))]π/40

8. Substitute the limits:

25/2*[(√(,2)*(1)+ln(√(,2)+1))−(1*(0)+ln(1+0))]

9. Simplify the final expression:

25/2*(√(,2)+ln(1+√(,2)))

Final Answer

(∫_0^5)(√(,x2+25)*d(x))=(25√(,2))/2+25/2*ln(1+√(,2))

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