Evaluate the Integral

Problem

(∫_^)((x3)/√(,x2+4)*d(x))

Solution

1. Identify the substitution method. Let u=x2+4 which implies x2=u−4

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

3. Rewrite the integral by splitting x3 into x2⋅x to match the substitution components.

(∫_^)((x2⋅x)/√(,x2+4)*d(x))

4. Substitute the variables u and d(u) into the integral.

(∫_^)((u−4)/√(,u)⋅1/2*d(u))

5. Distribute the constant and split the fraction into two terms.

1/2*(∫_^)((u1/2−4*u(−1/2))*d(u))

6. Integrate each term using the power rule.

1/2*(2/3*u3/2−8*u1/2)+C

7. Simplify the expression by distributing the 1/2

1/3*u3/2−4*u1/2+C

8. Back-substitute u=x2+4 to return to the original variable.

1/3*(x2+4)3/2−4*(x2+4)1/2+C

9. Factor out the common term (x2+4)1/2 to simplify the final form.

1/3*(x2+4)1/2*(x2+4−12)+C

1/3*(x2+4)1/2*(x2−8)+C

Final Answer

(∫_^)((x3)/√(,x2+4)*d(x))=((x2−8)√(,x2+4))/3+C

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