Evaluate the Integral

Problem

(∫_^)((ex)/√(,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)=(ex)2

2. Substitute u=ex to simplify the expression.

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

4. Rewrite the integral in terms of u by substituting ex*d(x) with d(u) and e(2*x) with u2

(∫_^)(1/√(,1−u2)*d(u))

5. Apply the formula for the integral of the inverse sine function, where (∫_^)(1/√(,1−u2)*d(u))=arcsin(u)+C

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

Final Answer

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

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