Evaluate the Integral

Problem

(∫_^)(x/√(,x−1)*d(x))

Solution

1. Identify the substitution to simplify the radicand. Let u=x−1

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

u=x−1

d(u)/d(x)=1

d(u)=d(x)

3. Solve for x in terms of u to substitute into the numerator.

x=u+1

4. Substitute the expressions for x √(,x−1) and d(x) into the integral.

(∫_^)((u+1)/√(,u)*d(u))

5. Distribute the denominator to split the integrand into two separate terms.

(∫_^)((u/√(,u)+1/√(,u))*d(u))

(∫_^)((u(1/2)+u(−1/2))*d(u))

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

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

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

7. Back-substitute u=x−1 to express the result in terms of x

2/3*(x−1)(3/2)+2*(x−1)(1/2)+C

Final Answer

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

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