Pregunta
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The integral evaluates to
.
Solución
Let
so that
When
, we have:
and when
, we have:
Thus, the integral becomes
Now, we evaluate the integral:
Applying the limits from
to
gives:
To combine the fractions, find a common denominator:
Thus, the value of the integral is
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Mind Expander
To evaluate the integral
, we can use the substitution
. Thus, we have:
- The differential change is given by
, which means . Now substitute and : - When
, . - When
, .
- When
So the limits of integration change from 0 and 1 to 5 and 6, respectively. The integral becomes:
- Now, we’ll compute this integral:
Evaluating this from 5 to 6 gives us:
To combine these fractions, find a common denominator, which is 360:
Now, we can subtract:
Therefore, the value of the definite integral
is:
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