Partial Fraction Integral 94D871

1. **State the problem:** Evaluate the integral $$\int \frac{7x^{2} + 8x - 4}{(x + 1)(x^{2} + x - 1)} \, dx$$. 2. **Factor and analyze the denominator:** The denominator is already factored as $$(x + 1)(x^{2} + x - 1)$$. 3. **Set up partial fraction decomposition:** We express the integrand as: $$\frac{7x^{2} + 8x - 4}{(x + 1)(x^{2} + x - 1)} = \frac{A}{x + 1} + \frac{Bx + C}{x^{2} + x - 1}$$ 4. **Multiply both sides by the denominator to clear fractions:** $$7x^{2} + 8x - 4 = A(x^{2} + x - 1) + (Bx + C)(x + 1)$$ 5. **Expand the right side:** $$A(x^{2} + x - 1) + (Bx + C)(x + 1) = A x^{2} + A x - A + B x^{2} + B x + C x + C$$ Combine like terms: $$= (A + B) x^{2} + (A + B + C) x + (C - A)$$ 6. **Equate coefficients from both sides:** For $x^{2}$: $$7 = A + B$$ For $x$: $$8 = A + B + C$$ For constant term: $$-4 = C - A$$ 7. **Solve the system:** From the first equation: $$B = 7 - A$$ From the third equation: $$C = -4 + A$$ Substitute $B$ and $C$ into the second equation: $$8 = A + (7 - A) + (-4 + A) = 7 + A - 4 = 3 + A$$ So, $$A = 5$$ Then, $$B = 7 - 5 = 2$$ $$C = -4 + 5 = 1$$ 8. **Rewrite the integral with partial fractions:** $$\int \frac{7x^{2} + 8x - 4}{(x + 1)(x^{2} + x - 1)} \, dx = \int \frac{5}{x + 1} \, dx + \int \frac{2x + 1}{x^{2} + x - 1} \, dx$$ 9. **Integrate the first term:** $$\int \frac{5}{x + 1} \, dx = 5 \ln|x + 1| + C_1$$ 10. **Integrate the second term:** Note that the denominator's derivative is: $$\frac{d}{dx}(x^{2} + x - 1) = 2x + 1$$ which matches the numerator exactly. Therefore, $$\int \frac{2x + 1}{x^{2} + x - 1} \, dx = \ln|x^{2} + x - 1| + C_2$$ 11. **Combine results:** $$\int \frac{7x^{2} + 8x - 4}{(x + 1)(x^{2} + x - 1)} \, dx = 5 \ln|x + 1| + \ln|x^{2} + x - 1| + C$$ where $C = C_1 + C_2$ is the constant of integration.