1. **State the problem:** Evaluate the definite integral $$\int_0^1 \frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 - 3x + 2}} \, dx$$. 2. **Simplify the denominator:** Factor the quadratic inside the square root: $$x^2 - 3x + 2 = (x-1)(x-2).$$ Since the integral is from 0 to 1, note that for $x \in [0,1]$, $(x-1) \leq 0$ and $(x-2) < 0$, so the product is positive and the square root is defined. 3. **Rewrite the denominator:** $$\sqrt{(x-1)(x-2)} = \sqrt{(1-x)(2-x)}$$ because both factors are negative in $[0,1]$, so we take the positive root of the positive product. 4. **Substitute to simplify the integral:** Let $$t = x - 1.5,$$ then $$x = t + 1.5,$$ and the limits change: - When $x=0$, $t = -1.5$. - When $x=1$, $t = -0.5$. Rewrite the denominator: $$\sqrt{(x-1)(x-2)} = \sqrt{t - 0.5)(t + 0.5)} = \sqrt{t^2 - 0.25}.$$ Rewrite numerator in terms of $t$: $$3x^3 - x^2 + 2x - 4 = 3(t+1.5)^3 - (t+1.5)^2 + 2(t+1.5) - 4.$$ Expand: - $(t+1.5)^3 = t^3 + 3 \cdot 1.5 t^2 + 3 \cdot (1.5)^2 t + (1.5)^3 = t^3 + 4.5 t^2 + 6.75 t + 3.375$ - $(t+1.5)^2 = t^2 + 3 t + 2.25$ So numerator: $$3(t^3 + 4.5 t^2 + 6.75 t + 3.375) - (t^2 + 3 t + 2.25) + 2(t + 1.5) - 4$$ $$= 3 t^3 + 13.5 t^2 + 20.25 t + 10.125 - t^2 - 3 t - 2.25 + 2 t + 3 - 4$$ $$= 3 t^3 + (13.5 - 1) t^2 + (20.25 - 3 + 2) t + (10.125 - 2.25 + 3 - 4)$$ $$= 3 t^3 + 12.5 t^2 + 19.25 t + 6.875.$$ 5. **Integral becomes:** $$\int_{-1.5}^{-0.5} \frac{3 t^3 + 12.5 t^2 + 19.25 t + 6.875}{\sqrt{t^2 - 0.25}} \, dt.$$ 6. **Use substitution:** Let $$t = \frac{1}{2} \sec \theta,$$ so that $$\sqrt{t^2 - \frac{1}{4}} = \sqrt{\frac{1}{4} \sec^2 \theta - \frac{1}{4}} = \frac{1}{2} \tan \theta.$$ Also, $$dt = \frac{1}{2} \sec \theta \tan \theta \, d\theta.$$ 7. **Change limits:** When $t = -1.5$, $$-1.5 = \frac{1}{2} \sec \theta \Rightarrow \sec \theta = -3 \Rightarrow \theta = \sec^{-1}(-3).$$ When $t = -0.5$, $$-0.5 = \frac{1}{2} \sec \theta \Rightarrow \sec \theta = -1 \Rightarrow \theta = \pi.$$ 8. **Rewrite integral in terms of $\theta$:** The integrand numerator becomes a polynomial in $t = \frac{1}{2} \sec \theta$. Calculate numerator: $$3 t^3 = 3 \left(\frac{1}{2} \sec \theta\right)^3 = 3 \frac{1}{8} \sec^3 \theta = \frac{3}{8} \sec^3 \theta,$$ $$12.5 t^2 = 12.5 \left(\frac{1}{2} \sec \theta\right)^2 = 12.5 \frac{1}{4} \sec^2 \theta = \frac{12.5}{4} \sec^2 \theta = 3.125 \sec^2 \theta,$$ $$19.25 t = 19.25 \frac{1}{2} \sec \theta = 9.625 \sec \theta,$$ $$6.875 = 6.875.$$ So numerator: $$\frac{3}{8} \sec^3 \theta + 3.125 \sec^2 \theta + 9.625 \sec \theta + 6.875.$$ Denominator: $$\sqrt{t^2 - \frac{1}{4}} = \frac{1}{2} \tan \theta.$$ Differential: $$dt = \frac{1}{2} \sec \theta \tan \theta \, d\theta.$$ 9. **Integral becomes:** $$\int_{\theta_1}^{\pi} \frac{\frac{3}{8} \sec^3 \theta + 3.125 \sec^2 \theta + 9.625 \sec \theta + 6.875}{\frac{1}{2} \tan \theta} \cdot \frac{1}{2} \sec \theta \tan \theta \, d\theta$$ Simplify: $$= \int_{\theta_1}^{\pi} \left(\frac{3}{8} \sec^3 \theta + 3.125 \sec^2 \theta + 9.625 \sec \theta + 6.875\right) \cdot \frac{\frac{1}{2} \sec \theta \tan \theta}{\frac{1}{2} \tan \theta} \, d\theta$$ $$= \int_{\theta_1}^{\pi} \left(\frac{3}{8} \sec^3 \theta + 3.125 \sec^2 \theta + 9.625 \sec \theta + 6.875\right) \sec \theta \, d\theta$$ $$= \int_{\theta_1}^{\pi} \left(\frac{3}{8} \sec^4 \theta + 3.125 \sec^3 \theta + 9.625 \sec^2 \theta + 6.875 \sec \theta\right) d\theta.$$ 10. **Evaluate the integral term-by-term:** Use known integrals for powers of secant: - $\int \sec \theta \, d\theta = \ln |\sec \theta + \tan \theta| + C$ - $\int \sec^2 \theta \, d\theta = \tan \theta + C$ - $\int \sec^3 \theta \, d\theta = \frac{1}{2} \sec \theta \tan \theta + \frac{1}{2} \ln |\sec \theta + \tan \theta| + C$ - $\int \sec^4 \theta \, d\theta = \tan \theta + \frac{1}{3} \sec^2 \theta \tan \theta + C$ (derived or from tables) Calculate each: $$\int \sec^4 \theta d\theta = \tan \theta + \frac{1}{3} \sec^2 \theta \tan \theta + C,$$ $$\int \sec^3 \theta d\theta = \frac{1}{2} \sec \theta \tan \theta + \frac{1}{2} \ln |\sec \theta + \tan \theta| + C,$$ $$\int \sec^2 \theta d\theta = \tan \theta + C,$$ $$\int \sec \theta d\theta = \ln |\sec \theta + \tan \theta| + C.$$ 11. **Combine with coefficients:** $$\int \left(\frac{3}{8} \sec^4 \theta + 3.125 \sec^3 \theta + 9.625 \sec^2 \theta + 6.875 \sec \theta\right) d\theta =$$ $$\frac{3}{8} \left(\tan \theta + \frac{1}{3} \sec^2 \theta \tan \theta\right) + 3.125 \left(\frac{1}{2} \sec \theta \tan \theta + \frac{1}{2} \ln |\sec \theta + \tan \theta|\right) + 9.625 \tan \theta + 6.875 \ln |\sec \theta + \tan \theta| + C.$$ Simplify coefficients: $$= \frac{3}{8} \tan \theta + \frac{3}{24} \sec^2 \theta \tan \theta + \frac{3.125}{2} \sec \theta \tan \theta + \frac{3.125}{2} \ln |\sec \theta + \tan \theta| + 9.625 \tan \theta + 6.875 \ln |\sec \theta + \tan \theta| + C.$$ $$= \left(\frac{3}{8} + 9.625\right) \tan \theta + \frac{1}{8} \sec^2 \theta \tan \theta + 1.5625 \sec \theta \tan \theta + \left(1.5625 + 6.875\right) \ln |\sec \theta + \tan \theta| + C.$$ $$= 9.875 \tan \theta + \frac{1}{8} \sec^2 \theta \tan \theta + 1.5625 \sec \theta \tan \theta + 8.4375 \ln |\sec \theta + \tan \theta| + C.$$ 12. **Evaluate at limits:** Recall $\theta_1 = \sec^{-1}(-3)$ and upper limit $\pi$. At $\theta = \pi$: - $\sec \pi = -1$, $\tan \pi = 0$, so terms with $\tan \pi$ vanish. - $\ln |\sec \pi + \tan \pi| = \ln |-1 + 0| = \ln 1 = 0$. So upper limit value is 0. At $\theta = \sec^{-1}(-3)$: - $\sec \theta = -3$. - $\tan \theta = -\sqrt{\sec^2 \theta - 1} = -\sqrt{9 - 1} = -\sqrt{8} = -2 \sqrt{2}$ (choosing negative because $\theta$ in second quadrant). - $\sec^2 \theta = 9$. - $\sec \theta + \tan \theta = -3 - 2 \sqrt{2}$, absolute value is $3 + 2 \sqrt{2}$. Calculate: $$9.875 \times (-2 \sqrt{2}) = -19.75 \sqrt{2},$$ $$\frac{1}{8} \times 9 \times (-2 \sqrt{2}) = \frac{9}{8} \times (-2 \sqrt{2}) = -\frac{18}{8} \sqrt{2} = -2.25 \sqrt{2},$$ $$1.5625 \times (-3) \times (-2 \sqrt{2}) = 1.5625 \times 6 \sqrt{2} = 9.375 \sqrt{2},$$ $$8.4375 \ln(3 + 2 \sqrt{2}).$$ Sum these: $$-19.75 \sqrt{2} - 2.25 \sqrt{2} + 9.375 \sqrt{2} + 8.4375 \ln(3 + 2 \sqrt{2}) = (-19.75 - 2.25 + 9.375) \sqrt{2} + 8.4375 \ln(3 + 2 \sqrt{2})$$ $$= (-12.625) \sqrt{2} + 8.4375 \ln(3 + 2 \sqrt{2}).$$ 13. **Final value of integral:** $$0 - \left((-12.625) \sqrt{2} + 8.4375 \ln(3 + 2 \sqrt{2})\right) = 12.625 \sqrt{2} - 8.4375 \ln(3 + 2 \sqrt{2}).$$ **Answer:** $$\boxed{12.625 \sqrt{2} - 8.4375 \ln(3 + 2 \sqrt{2})}.$$