Evaluate the Integral

Problem

(∫_^)((ex+e(−x))/(ex−e(−x))*d(x))

Solution

1. Identify the structure of the integrand and notice that the numerator is the derivative of the denominator.

2. Substitute u=ex−e(−x) to simplify the integral.

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=ex−(−e(−x))

d(u)/d(x)=ex+e(−x)

d(u)=(ex+e(−x))*d(x)

4. Rewrite the integral in terms of u

(∫_^)(1/u*d(u))

5. Integrate using the natural logarithm rule.

ln(u)+C

6. Back-substitute the original expression for u

ln(ex−e(−x))+C

Final Answer

(∫_^)((ex+e(−x))/(ex−e(−x))*d(x))=ln(ex−e(−x))+C

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