Evaluate the Integral

Problem

(∫_^)(e^x*cos(e^x)*d(x))

Solution

1. Identify the inner function for substitution, which is u=e^x

2. Differentiate u with respect to x to find d(u)=e^x*d(x)

3. Substitute u and d(u) into the integral to rewrite it in terms of u

(∫_^)(cos(u)*d(u))

4. Integrate the cosine function with respect to u

sin(u)+C

5. Back-substitute the original expression e^x for u to get the final result.

Final Answer

(∫_^)(e^x*cos(e^x)*d(x))=sin(e^x)+C

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