Evaluate the Integral

Problem

(∫_^)((−x)/((x+1)−√(,x+1))*d(x))

Solution

1. Substitute to simplify the radical by letting u=√(,x+1)

2. Square both sides to find u^2=x+1 which implies x=u^2−1

3. Differentiate the substitution to find d(x)=2*u*d(u)

4. Substitute these expressions into the integral:

(∫_^)((−(u^2−1))/(u^2−u)⋅2*u*d(u))

5. Simplify the integrand by factoring the denominator u^2−u=u*(u−1) and the numerator u^2−1=(u−1)*(u+1)

(∫_^)((−(u−1)*(u+1))/(u*(u−1))⋅2*u*d(u))

6. Cancel the common terms (u−1) and u

(∫_^)(−2*(u+1)*d(u))

7. Integrate the resulting polynomial:

−2*(∫_^)((u+1)*d(u))=−2*((u^2)/2+u)+C

8. Distribute the constant:

−u^2−2*u+C

9. Back-substitute u=√(,x+1) and u^2=x+1 to return to the variable x

−(x+1)−2√(,x+1)+C

10. Combine the constant −1 into the arbitrary constant C

−x−2√(,x+1)+C

Final Answer

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

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