Evaluate the Integral

Problem

(∫_^)((x3)/√(,x2+36)*d(x))

Solution

1. Identify the substitution method. Let u=x2+36

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

3. Express the remaining x2 term in terms of u using x2=u−36

4. Substitute these into the integral to rewrite it in terms of u

(∫_^)((x2⋅x)/√(,x2+36)*d(x))=(∫_^)((u−36)/√(,u)⋅1/2*d(u))

5. Distribute the fraction and simplify the integrand.

1/2*(∫_^)((u(1/2)−36*u(−1/2))*d(u))

6. Integrate with respect to u using the power rule.

1/2*(2/3*u(3/2)−72*u(1/2))+C

7. Simplify the expression by distributing the constant.

1/3*u(3/2)−36*u(1/2)+C

8. Back-substitute u=x2+36 to return to the original variable.

1/3*(x2+36)(3/2)−36*(x2+36)(1/2)+C

9. Factor out the common term (x2+36)(1/2) to simplify the final form.

1/3*(x2+36)(1/2)*(x2+36−108)+C

1/3*(x2+36)(1/2)*(x2−72)+C

Final Answer

(∫_^)((x3)/√(,x2+36)*d(x))=((x2−72)√(,x2+36))/3+C

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