Evaluate the Integral

Problem

(∫_0^1)((r^3)/√(,16+r^2)*d(r))

Solution

1. Substitute to simplify the radical by letting u=16+r^2

2. Differentiate the substitution to find d(u)=2*r*d(r) which implies r*d(r)=1/2*d(u)

3. Rewrite the numerator r^3 as r^2⋅r From our substitution, r^2=u−16

4. Change the limits of integration: when r=0 u=16+0^2=16 when r=1 u=16+1^2=17

5. Transform the integral into terms of u

(∫_16^17)((u−16)/√(,u)⋅1/2*d(u))

6. Distribute the denominator and simplify the integrand:

1/2*(∫_16^17)((u^(1/2)−16*u^(−1/2))*d(u))

7. Integrate using the power rule:

1/2*[2/3*u^(3/2)−32*u^(1/2)]^17_16

8. Evaluate at the upper and lower limits:

1/3*(17)^(3/2)−16*(17)^(1/2)−(1/3*(16)^(3/2)−16*(16)^(1/2))

9. Simplify the numerical values:

(17√(,17))/3−16√(,17)−(64/3−64)

10. Combine like terms:

(17√(,17)−48√(,17))/3−((64−192)/3)

(−31√(,17))/3+128/3

Final Answer

(∫_0^1)((r^3)/√(,16+r^2)*d(r))=(128−31√(,17))/3

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