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Answer
The integral evaluates to
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Mind Expander
To solve the integral
we can begin by breaking down the fraction into partial fractions. We express the integrand as:
where
Simplifying this gives us:
Now, we can match coefficients with the polynomial on the right:
- For
: There is no term, so its coefficient must be , - For
: The coefficient must be , - For
: The coefficient must be , - For
: The coefficient must be , - For the constant term: It must be
.
From this, we can set up a system of equations:
- Coefficient of
: (no restrictions) - Coefficient of
: (no restrictions) - Coefficient of
: , - Coefficient of
: , - Constant term:
.
From
Now we have
Next, we can integrate term by term:
The first integral is straightforward:
For the second integral, we can break it down further:
The integral of the first part can be solved using substitution
The second integral is simply:
Putting it all together, we have:
where
Thus, the final answer is:
