Evaluate the Integral

Problem

(∫_^)(1/(1+√(,x))*d(x))

Solution

1. Substitute a new variable to eliminate the square root by letting u=√(,x)

2. Differentiate the substitution equation x=u2 to find d(x)=2*u*d(u)

3. Rewrite the integral in terms of u by substituting the expressions for √(,x) and d(x)

(∫_^)((2*u)/(1+u)*d(u))

4. Perform polynomial division or rewrite the numerator as 2*u=2*(u+1)−2 to simplify the integrand.

(∫_^)((2*(u+1)−2)/(u+1)*d(u))

(∫_^)((2−2/(u+1))*d(u))

5. Integrate each term with respect to u

2*u−2*ln(u+1)+C

6. Back-substitute u=√(,x) to return to the original variable.

Final Answer

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

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