Evaluate the Integral

Problem

(∫_^)(1/(√(,x)*(1+√(,x)))*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, choosing u to be the expression inside the parentheses.

u=1+√(,x)

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

d(u)/d(x)=1/(2√(,x))

3. Rearrange the differential equation to solve for the terms present in the integral.

2*d(u)=1/√(,x)*d(x)

4. Substitute the expressions for u and d(u) into the original integral.

(∫_^)(2/u*d(u))

5. Integrate the simplified expression using the natural logarithm rule.

2*ln(u)+C

6. Back-substitute the original expression for u to get the final result in terms of x

2*ln(1+√(,x))+C

Final Answer

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

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