Question
Studdy Solution
STEP 1
1. The given problem involves evaluating a double integral.
2. The integrals are with respect to different variables, suggesting a potential separation of variables.
3. We assume is treated as a constant when integrating with respect to , and vice versa.
STEP 2
1. Evaluate the integral with respect to .
2. Evaluate the integral with respect to .
3. Combine the results.
STEP 3
Evaluate the integral with respect to :
To solve this, we integrate term by term:
- For , use integration by parts.
- For , use integration by parts.
- For , recognize it as a simple exponential integral.
STEP 4
Perform integration by parts for the first term :
Let and .
Then, and .
Apply integration by parts:
STEP 5
Perform integration by parts for the remaining integral from the previous step:
Let and .
Then, and .
Apply integration by parts:
Substitute back:
STEP 6
Evaluate the integral :
This is similar to the previous integration by parts:
STEP 7
Evaluate the integral :
This is a simple exponential integral:
STEP 8
Combine the results of the integrals with respect to :
STEP 9
Evaluate the integral with respect to :
This can be split into two separate integrals:
STEP 10
Evaluate :
Since is independent of , it can be treated as a constant:
STEP 11
Evaluate :
STEP 12
Combine the results of the integrals with respect to :
STEP 13
Combine the results from both integrals:
The final result is:
Simplify if possible:
The solution to the problem is:
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