Evaluate the Integral

Problem

(∫_^)(x3√(,x2−1)*d(x))

Solution

1. Identify the substitution to simplify the integrand. Let u=x2−1

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

d(u)=2*x*d(x)

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

3. Rewrite the expression for x2 in terms of u

x2=u+1

4. Substitute the expressions into the integral.

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

5. Distribute the √(,u) (which is u(1/2) across the terms in the parentheses.

1/2*(∫_^)((u(3/2)+u(1/2))*d(u))

6. Integrate each term using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

1/2*(2/5*u(5/2)+2/3*u(3/2))+C

7. Simplify the expression by distributing the 1/2

1/5*u(5/2)+1/3*u(3/2)+C

8. Back-substitute u=x2−1 to return to the original variable.

1/5*(x2−1)(5/2)+1/3*(x2−1)(3/2)+C

Final Answer

(∫_^)(x3√(,x2−1)*d(x))=1/5*(x2−1)(5/2)+1/3*(x2−1)(3/2)+C

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