Integration of 1/e^y

To find the integration of 1/e^y (e to the power y), we can use the substitution method. We substitute a variable in place of the exponent and then differentiate it to find the value of the original expression. Let's break down the process step by step:

Step 1: Identifying the Substitution Variable
In this case, we can let u = e^y be our substitution variable. This choice is beneficial because when we differentiate u with respect to y, we get du/dy = e^y, which matches the denominator of our original expression.

Step 2: Differentiating the Substitution Variable
Taking the derivative of u = e^y with respect to y, we get du/dy = e^y.

Step 3: Rearranging the Integration Expression
Now, we can rewrite our original expression in terms of u:
∫(1/e^y) dy = ∫(1/u) du

Step 4: Evaluating the Integration Expression
The integral of 1/u with respect to u is ln|u| + C, where C is the constant of integration. Therefore, the integration of 1/e^y is:
∫(1/e^y) dy = ln|u| + C = ln|e^y| + C = ln(e^y) + C

Step 5: Simplifying the Final Result
Using the property of logarithms, ln(e^y) simplifies to y. Therefore, the final result of the integration is:
∫(1/e^y) dy = ln(e^y) + C = y + C

Conclusion
The integration of 1/e^y is equal to y plus a constant of integration. This means that the original expression can be represented as y + C, where C is an arbitrary constant.