1. **Evaluate** $\int_0^2 (3x^2 - 2x + 1) \, dx$. Formula: $\int (ax^n) dx = \frac{a}{n+1} x^{n+1} + C$. Calculate: $$\int_0^2 3x^2 dx = \left[x^3\right]_0^2 = 8$$ $$\int_0^2 -2x dx = \left[-x^2\right]_0^2 = -4$$ $$\int_0^2 1 dx = \left[x\right]_0^2 = 2$$ Sum: $8 - 4 + 2 = 6$. 2. **Evaluate** $\int (x^{1/2} + 4x^3) dx$. Use power rule: $$\int x^{1/2} dx = \frac{2}{3} x^{3/2} + C$$ $$\int 4x^3 dx = x^4 + C$$ Result: $\frac{2}{3} x^{3/2} + x^4 + C$. 3. **Evaluate** $\int_1^4 (\sqrt{x} + \frac{1}{\sqrt{x}}) dx$. Rewrite: $$\int_1^4 (x^{1/2} + x^{-1/2}) dx = \left[ \frac{2}{3} x^{3/2} + 2 x^{1/2} \right]_1^4$$ Calculate: At 4: $\frac{2}{3} \cdot 8 + 2 \cdot 2 = \frac{16}{3} + 4 = \frac{28}{3}$ At 1: $\frac{2}{3} + 2 = \frac{8}{3}$ Difference: $\frac{28}{3} - \frac{8}{3} = \frac{20}{3}$. 4. **Evaluate** $\int_0^\pi \cos x \, dx$. Integral: $$\sin x \Big|_0^\pi = \sin \pi - \sin 0 = 0 - 0 = 0$$. 5. **Evaluate** $\int (e^x + \sin x) dx$. Integrate termwise: $$\int e^x dx = e^x + C$$ $$\int \sin x dx = -\cos x + C$$ Result: $e^x - \cos x + C$. 6. **Evaluate** $\int_0^1 (2x + 1)^3 dx$. Expand: $$(2x+1)^3 = 8x^3 + 12x^2 + 6x + 1$$ Integrate termwise: $$\int_0^1 8x^3 dx = 2$$ $$\int_0^1 12x^2 dx = 4$$ $$\int_0^1 6x dx = 3$$ $$\int_0^1 1 dx = 1$$ Sum: $2 + 4 + 3 + 1 = 10$. 7. **Evaluate** $\int \frac{\cos(\sqrt{x})}{\sqrt{x}} dx$. Substitute $t = \sqrt{x}$, so $x = t^2$, $dx = 2t dt$. Rewrite integral: $$\int \frac{\cos t}{t} 2t dt = 2 \int \cos t dt = 2 \sin t + C = 2 \sin(\sqrt{x}) + C$$. 8. **Evaluate** $\int_0^2 |x - 1| dx$. Split at $x=1$: $$\int_0^1 (1 - x) dx + \int_1^2 (x - 1) dx$$ Calculate: $$\left[x - \frac{x^2}{2}\right]_0^1 = 1 - \frac{1}{2} = \frac{1}{2}$$ $$\left[\frac{x^2}{2} - x\right]_1^2 = \left(2 - 2\right) - \left(\frac{1}{2} - 1\right) = 0 - (-\frac{1}{2}) = \frac{1}{2}$$ Sum: $\frac{1}{2} + \frac{1}{2} = 1$. 9. **Evaluate** $\int \frac{x}{x^2 + 1} dx$. Substitute $u = x^2 + 1$, $du = 2x dx$. Rewrite: $$\frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln|u| + C = \frac{1}{2} \ln(x^2 + 1) + C$$. 10. **Evaluate** $\int_0^{\pi/4} \sec^2 \theta d\theta$. Integral: $$\tan \theta \Big|_0^{\pi/4} = 1 - 0 = 1$$. 11. **Evaluate** $\int_1^e \frac{\ln x}{x} dx$. Substitute $t = \ln x$, $dt = \frac{1}{x} dx$. Integral becomes: $$\int_0^1 t dt = \frac{1}{2} t^2 \Big|_0^1 = \frac{1}{2}$$. 12. **Evaluate** $\int \sin(3x) dx$. Integral: $$-\frac{1}{3} \cos(3x) + C$$. 13. **Evaluate** $\int_{-1}^2 (e^x + 2) dx$. Split: $$\int_{-1}^2 e^x dx + \int_{-1}^2 2 dx = \left[e^x\right]_{-1}^2 + 2(x)\Big|_{-1}^2 = (e^2 - e^{-1}) + 2(3) = e^2 - \frac{1}{e} + 6$$. 14. **Evaluate** $\int x e^{x^2} dx$. Substitute $u = x^2$, $du = 2x dx$. Rewrite: $$\frac{1}{2} \int e^u du = \frac{1}{2} e^{x^2} + C$$. 15. **Evaluate** $\int_0^1 \frac{dx}{\sqrt{1 - x^2}}$. Integral is arcsin: $$\arcsin x \Big|_0^1 = \frac{\pi}{2} - 0 = \frac{\pi}{2}$$. 16. **Evaluate** $\int \frac{(\arctan x)^2}{1 + x^2} dx$. Substitute $t = \arctan x$, $dt = \frac{1}{1+x^2} dx$. Integral becomes: $$\int t^2 dt = \frac{t^3}{3} + C = \frac{(\arctan x)^3}{3} + C$$. 17. **Evaluate** $\int_0^{\pi/2} \sin x \cos x dx$. Use identity: $$\sin x \cos x = \frac{1}{2} \sin(2x)$$ Integral: $$\frac{1}{2} \int_0^{\pi/2} \sin(2x) dx = \frac{1}{2} \left[-\frac{\cos(2x)}{2}\right]_0^{\pi/2} = -\frac{1}{4} (\cos \pi - \cos 0) = -\frac{1}{4} (-1 - 1) = \frac{1}{2}$$. 18. **Evaluate** $\int \frac{2t + 3}{t^2 + 3t + 5} dt$. Derivative of denominator: $$\frac{d}{dt} (t^2 + 3t + 5) = 2t + 3$$ Integral: $$\int \frac{f'(t)}{f(t)} dt = \ln|t^2 + 3t + 5| + C$$. 19. **Evaluate** $\int_0^4 (4 - t) \sqrt{t} dt$. Rewrite: $$\int_0^4 (4 \sqrt{t} - t \sqrt{t}) dt = \int_0^4 (4 t^{1/2} - t^{3/2}) dt$$ Integrate: $$4 \cdot \frac{2}{3} t^{3/2} - \frac{2}{5} t^{5/2} \Big|_0^4 = \frac{8}{3} \cdot 8 - \frac{2}{5} \cdot 32 = \frac{64}{3} - \frac{64}{5} = \frac{320 - 192}{15} = \frac{128}{15}$$. 20. **Evaluate** $\int_{-2}^2 f(x) dx$ where $$f(x) = \begin{cases} 3 & -2 \leq x \leq 0 \\ 4 - x^2 & 0 < x \leq 2 \end{cases}$$ Split integral: $$\int_{-2}^0 3 dx + \int_0^2 (4 - x^2) dx = 3(0 + 2) + \left[4x - \frac{x^3}{3}\right]_0^2 = 6 + (8 - \frac{8}{3}) = 6 + \frac{16}{3} = \frac{34}{3}$$. **Final answers:** 1. 6 2. $\frac{2}{3} x^{3/2} + x^4 + C$ 3. $\frac{20}{3}$ 4. 0 5. $e^x - \cos x + C$ 6. 10 7. $2 \sin(\sqrt{x}) + C$ 8. 1 9. $\frac{1}{2} \ln(x^2 + 1) + C$ 10. 1 11. $\frac{1}{2}$ 12. $-\frac{1}{3} \cos(3x) + C$ 13. $e^2 - \frac{1}{e} + 6$ 14. $\frac{1}{2} e^{x^2} + C$ 15. $\frac{\pi}{2}$ 16. $\frac{(\arctan x)^3}{3} + C$ 17. $\frac{1}{2}$ 18. $\ln|t^2 + 3t + 5| + C$ 19. $\frac{128}{15}$ 20. $\frac{34}{3}$