Question
Use partial fractions to integrate the following:
Ask by Harmon Reid. in South Africa
Mar 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The integral of
Solution
We wish to evaluate
Step 1. Factorize the Denominator
Notice that
Thus, the integrand can be rewritten as
Step 2. Decompose into Partial Fractions
We set
Multiply both sides by
Expanding the right-hand side:
Then,
Step 3. Equate Coefficients
Match the coefficients of like powers of
- Coefficient of
: - Coefficient of
: - Constant term:
From
Then using
Thus, the partial fractions are
Step 4. Integrate Each Term
Now, integrate term-by-term:
Step 5. Combine the Results
Thus, the integral becomes
where
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Simplify this solution Bonus Knowledge
To solve the integral using partial fractions, we first factor the denominator:
Now, we set up the partial fraction decomposition:
Multiplying through by the common denominator
Expanding the right-hand side:
Next, we gather coefficients for
-
(coefficient of ), -
(coefficient of ), -
(constant term).
From
Thus, we have:
Now integrating each term:
And there you go! The integral is:
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Every time someone encounters a complex fraction, they might feel a slight panic. Common mistakes include forgetting to properly factor the denominator or miscalculating the coefficients during the setup of partial fractions. Take a moment to double-check your factorizations and your algebra—your future self will thank you for it!
