Evaluate the Integral integral of e^(1-x) with respect to x

Problem

(∫_^)(e(1−x)*d(x))

Solution

1. Identify the form of the integral, which is an exponential function with a linear exponent.

2. Apply the substitution method by letting u=1−x

3. Differentiate u to find d(u)=−d(x) which implies d(x)=−d(u)

4. Substitute the variables into the integral to get (∫_^)(−eu*d(u))

5. Integrate the exponential function, resulting in −eu+C

6. Back-substitute the original expression for u to obtain the final result.

Final Answer

(∫_^)(e(1−x)*d(x))=−e(1−x)+C

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