Evaluate the Integral

Problem

(∫_^)((e^x)/√(,1−e^(2*x))*d(x))

Solution

1. Identify the structure of the integrand as a potential candidate for an inverse trigonometric substitution, noting that e^(2*x)=(e^x)^2

2. Substitute u=e^x to simplify the expression.

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

4. Rewrite the integral in terms of u by substituting e^x*d(x) with d(u) and e^(2*x) with u^2

(∫_^)(1/√(,1−u^2)*d(u))

5. Apply the formula for the integral of the inverse sine function, where (∫_^)(1/√(,1−u^2)*d(u))=arcsin(u)+C

6. Back-substitute the original expression for u to return to the variable x

Final Answer

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

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