Make Tough INTEGRATION Easy | PARTIAL FRACTIONS

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

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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