Calculate the following integral ∫ ∫ 𝑒𝑦/𝑥 𝑑𝑥 𝑑𝑦

To calculate the given integral ∫ ∫ 𝑒𝑦/𝑥 𝑑𝑥 𝑑𝑦, we will use the method of iterated integration.

Step 1: Integrate with respect to x: ∫ 𝑒𝑦/𝑥 𝑑𝑥 = 𝑒𝑦 ∫ 1/𝑥 𝑑𝑥

Step 2: Evaluate the integral with respect to x: ∫ 1/𝑥 𝑑𝑥 = ln|𝑥|

Step 3: Substitute the result back into the original integral: ∫ ∫ 𝑒𝑦/𝑥 𝑑𝑥 𝑑𝑦 = ∫ 𝑒𝑦 ln|𝑥| 𝑑𝑦

Step 4: Integrate with respect to y: ∫ 𝑒𝑦 ln|𝑥| 𝑑𝑦 = ln|𝑥| ∫ 𝑒𝑦 𝑑𝑦

Step 5: Evaluate the integral with respect to y: ∫ 𝑒𝑦 𝑑𝑦 = 𝑒𝑦

Step 6: Substitute the result back into the original integral: ∫ ∫ 𝑒𝑦/𝑥 𝑑𝑥 𝑑𝑦 = ln|𝑥| 𝑒𝑦 + 𝐶

Therefore, the final result of the integral is ln|𝑥| 𝑒𝑦 + 𝐶, where 𝐶 is the constant of integration.