Evaluate the Integral

Problem

(∫_0^1)((r3)/√(,4+r2)*d(r))

Solution

1. Identify a suitable substitution to simplify the integrand. Let u=4+r2

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

d(u)/d(r)=2*r

r*d(r)=1/2*d(u)

3. Rewrite the numerator r3 as r2⋅r and express r2 in terms of u

r2=u−4

4. Change the limits of integration from r to u

When *r=0,u=4+0=4

When *r=1,u=4+1=5

5. Substitute all components into the integral.

(∫_4^5)((u−4)/√(,u)⋅1/2*d(u))

6. Distribute the fraction and simplify the integrand.

1/2*(∫_4^5)((u(1/2)−4*u(−1/2))*d(u))

7. Integrate using the power rule.

1/2*[2/3*u(3/2)−8*u(1/2)]54

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

1/2*[(2/3*(5√(,5))−8√(,5))−(2/3*(8)−8*(2))]

9. Simplify the numerical values.

1/2*[((10√(,5))/3−(24√(,5))/3)−(16/3−48/3)]

1/2*[−(14√(,5))/3+32/3]

(16−7√(,5))/3

Final Answer

(∫_0^1)((r3)/√(,4+r2)*d(r))=(16−7√(,5))/3

Want more problems? Check here!