Скачать или смотреть Using PARTIAL FRACTIONS to INTEGRATE an EXPRESSION

In this question we've been asked to rewrite the expression given to using partial fractions, before integrating it.

Audio correction at 06:20 - "So, -4 is equal to 4A".

Key Steps

1. Set up the decomposition
2. Equate numerators
3. Solve for B (by eliminating A)
4. Solve for A (by eliminating B)
5. Write the partial fractions
6. Integrate the sum (linearity)
7. Apply the Natural Log Rule
8. Simplifying using logarithm rules

Key Points

1. Partial Fraction Decomposition: A method to decompose a complex fraction into simpler fractions for easier integration.
2. Decomposition Process: Introduce unknown constants (A and B), equate numerators after finding a common denominator, and solve for A and B by substituting values of x.
3. Integration: Integrate the decomposed partial fractions and simplify the solution.
4. Partial Fraction Decomposition: The given expression is decomposed into partial fractions: 3/(x+3) - 1/(x-1).
5. Integration Process: Each term is integrated separately using the Natural Log Rule, resulting in 3ln|x+3| - ln|x-1| + c.
6. Simplified Result: Applying logarithmic rules, the final answer is ln| (x+3)^3 / (x-1) | + c.

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