Question
Ask by Wilson Bryant. in Tunisia
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The integral simplifies to
.
Solution
To evaluate the integral
we can use integration by parts. Integration by parts is based on the formula:
where
and
are continuously differentiable functions on the interval of integration.
Step 1: Choose
and
Let’s assign:
Step 2: Apply Integration by Parts
Substituting into the integration by parts formula:
Step 3: Evaluate the Boundary Terms
Compute the boundary term
:
Step 4: Combine the Results
Putting it all together:
Final Result
Thus, the integral simplifies to:
This result expresses the original integral in terms of the function
evaluated at the upper limit and the integral of
itself over the interval
.
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Mind Expander
Did you know that integration by parts can be a real lifesaver when tackling integrals like this? For your integral
, you can apply the integration by parts formula
. In this case, let
and
. This method not only simplifies the integral but gives you an insightful result related to
itself!
Another fun fact is that this integral can often be interpreted in the context of physics or economics. For instance, if
represents a cost function, then
might represent the total cost associated with producing a certain quantity over the interval from 0 to 1. This connection between mathematical concepts and real-world situations really illuminates the power of calculus in understanding and solving practical problems we encounter every day!