Evaluate the Integral

Problem

(∫_^)(x^3√(,x^2−1)*d(x))

Solution

1. Identify the substitution to simplify the integrand. Let u=x^2−1

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 x^2 in terms of u

x^2=u+1

4. Substitute the expressions into the integral.

(∫_^)(x^2√(,x^2−1)⋅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 (∫_^)(u^n*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=x^2−1 to return to the original variable.

1/5*(x^2−1)^(5/2)+1/3*(x^2−1)^(3/2)+C

Final Answer

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

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