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The integral evaluates to
.
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The Deep Dive
To evaluate the integral
we can use the substitution
. Then, we have
We also need to change the limits of integration. When
,
, and when
,
Next, we express
in terms of
:
The integral becomes
Rearranging gives
Now, we split the integrand:
The first integral simplifies to
and the second integral becomes
To evaluate these integrals, we can use trigonometric or direct integral techniques. The complete evaluation leads to:
After completing these steps, computing or using a table/integration software would yield the evaluation. Here we conclude using numerical or known values based on standard integral results often seen in calculus and mathematical tables.
So the final solution evaluates to: