Evaluate the Integral

Problem

(∫_^)(1/(√(,x)*(1−√(,x)))*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, choosing u=1−√(,x)

2. Differentiate the substitution 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 d(x)

d(x)=−2√(,x)*d(u)

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

(∫_^)(1/(√(,x)⋅u)⋅(−2√(,x))*d(u))

5. Simplify the integrand by canceling the √(,x) terms.

−2*(∫_^)(1/u*d(u))

6. Integrate with respect to u using the natural logarithm rule.

−2*ln(u)+C

7. 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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