1. Problem (a): Evaluate $\int (1 - \frac{1}{w})\cos(w - \ln(w))\, dw$. Step 1: Let $u = w - \ln(w)$. Then, compute $du$. $$du = \frac{d}{dw}(w) - \frac{d}{dw}(\ln w) = 1 - \frac{1}{w}$$ Step 2: Substitute in the integral: $$\int (1 - \frac{1}{w}) \cos(w - \ln w) dw = \int \cos(u) du$$ Step 3: Integrate: $$\int \cos(u) du = \sin(u) + C = \sin(w - \ln w) + C$$ --- 2. Problem (b): Evaluate $\int 3(8y - 1) e^{4y^2 - y} dy$. Step 1: Let $v = 4y^2 - y$, then compute $dv$. $$dv = 8y - 1 dy$$ Step 2: Rewrite the integral: $$\int 3(8y - 1) e^{4y^2 - y} dy = 3 \int (8y - 1) e^v dy$$ Since $dv = (8y - 1) dy$, then $(8y - 1) dy = dv$, so $$= 3 \int e^v dv = 3 e^v + C = 3 e^{4y^2 - y} + C$$ --- 3. Problem (c): Evaluate $\int x^2 (3 - 10x^3)^4 dx$. Step 1: Let $t = 3 - 10x^3$. Compute $dt$: $$dt = -30 x^2 dx \implies x^2 dx = -\frac{dt}{30}$$ Step 2: Substitute in the integral: $$\int x^2 (3 - 10x^3)^4 dx = \int t^4 \left(-\frac{dt}{30}\right) = -\frac{1}{30} \int t^4 dt$$ Step 3: Integrate: $$-\frac{1}{30} \cdot \frac{t^5}{5} + C = -\frac{t^5}{150} + C = -\frac{(3 - 10x^3)^5}{150} + C$$ --- 4. Problem (d): Evaluate $\int \frac{x}{\sqrt{1 - 4x^2}} dx$. Step 1: Let $w = 1 - 4x^2$. Compute $dw$: $$dw = -8x dx \implies x dx = -\frac{dw}{8}$$ Step 2: Substitute in the integral: $$\int \frac{x}{\sqrt{1 - 4x^2}} dx = \int \frac{x}{\sqrt{w}} dx$$ Express $x dx$: $$x dx = -\frac{dw}{8}$$ So the integral becomes: $$\int \frac{x}{\sqrt{w}} dx = \int \frac{-\frac{dw}{8}}{\sqrt{w}} = -\frac{1}{8} \int w^{-1/2} dw$$ Step 3: Integrate: $$-\frac{1}{8} \cdot 2 w^{1/2} + C = -\frac{1}{4} \sqrt{w} + C = -\frac{1}{4} \sqrt{1 - 4x^2} + C$$ --- 5. Geometry problem: Triangle with vertices A(0,0), B(4,0), and C(4,6). To find hypotenuse length (side AC): $$\text{length} = \sqrt{(4-0)^2 + (6-0)^2} = \sqrt{16 + 36} = \sqrt{52}$$ Simplify: $$\sqrt{52} = 2\sqrt{13}$$ --- Final answers: (a) $\sin(w - \ln w) + C$ (b) $3 e^{4y^{2} - y} + C$ (c) $-\frac{(3 - 10x^{3})^5}{150} + C$ (d) $-\frac{1}{4} \sqrt{1 - 4x^{2}} + C$ Triangle hypotenuse length: $2\sqrt{13}$