1. Problem: Evaluate the integral $$\int \frac{5}{4 - \sqrt{3} - z} \, dz$$ Step 1: Recognize this is an integral of the form $$\int \frac{c}{a - z} \, dz$$ where $$c=5$$ and $$a=4 - \sqrt{3}$$. Step 2: The integral formula is $$\int \frac{c}{a - z} \, dz = -c \ln|a - z| + C$$ where $$C$$ is the constant of integration. Step 3: Applying the formula: $$\int \frac{5}{4 - \sqrt{3} - z} \, dz = -5 \ln|4 - \sqrt{3} - z| + C$$ Alternative form: $$5 \ln|z - (4 - \sqrt{3})| + C$$ (using $$\ln|a - z| = \ln|z - a|$$ and changing sign accordingly). 2. Problem: Evaluate the integral $$\int \frac{w}{w - 2\sqrt{6} - 3w - 5} \, dw$$ Step 1: Simplify the denominator: $$w - 2\sqrt{6} - 3w - 5 = -2w - 2\sqrt{6} - 5$$ Step 2: Rewrite the integral: $$\int \frac{w}{-2w - 2\sqrt{6} - 5} \, dw = -\int \frac{w}{2w + 2\sqrt{6} + 5} \, dw$$ Step 3: Use substitution: Let $$u = 2w + 2\sqrt{6} + 5$$, then $$du = 2 dw$$, so $$dw = \frac{du}{2}$$. Step 4: Express $$w$$ in terms of $$u$$: $$u = 2w + 2\sqrt{6} + 5 \Rightarrow w = \frac{u - 2\sqrt{6} - 5}{2}$$ Step 5: Substitute into integral: $$-\int \frac{\frac{u - 2\sqrt{6} - 5}{2}}{u} \cdot \frac{du}{2} = -\int \frac{u - 2\sqrt{6} - 5}{4u} du = -\frac{1}{4} \int \left(1 - \frac{2\sqrt{6} + 5}{u}\right) du$$ Step 6: Integrate term by term: $$-\frac{1}{4} \left( \int 1 \, du - (2\sqrt{6} + 5) \int \frac{1}{u} \, du \right) = -\frac{1}{4} \left( u - (2\sqrt{6} + 5) \ln|u| \right) + C$$ Step 7: Substitute back $$u$$: $$= -\frac{1}{4} \left( 2w + 2\sqrt{6} + 5 - (2\sqrt{6} + 5) \ln|2w + 2\sqrt{6} + 5| \right) + C$$ Alternative form: $$= -\frac{1}{2} w - \frac{2\sqrt{6} + 5}{4} + \frac{2\sqrt{6} + 5}{4} \ln|2w + 2\sqrt{6} + 5| + C$$ 3. Problem: Evaluate the integral $$\int \cos(\sqrt{x}) \, dx$$ Step 1: Use substitution: Let $$t = \sqrt{x} = x^{1/2}$$, then $$x = t^2$$ and $$dx = 2t \, dt$$. Step 2: Rewrite integral: $$\int \cos(t) \cdot 2t \, dt = 2 \int t \cos(t) \, dt$$ Step 3: Use integration by parts: Let $$u = t$$, $$dv = \cos(t) dt$$ Then $$du = dt$$, $$v = \sin(t)$$ Step 4: Apply integration by parts formula: $$\int u \, dv = uv - \int v \, du$$ Step 5: Compute: $$2 \int t \cos(t) dt = 2 (t \sin(t) - \int \sin(t) dt) = 2 (t \sin(t) + \cos(t)) + C$$ Step 6: Substitute back $$t = \sqrt{x}$$: $$= 2 \sqrt{x} \sin(\sqrt{x}) + 2 \cos(\sqrt{x}) + C$$ Alternative form: $$2 (\sqrt{x} \sin(\sqrt{x}) + \cos(\sqrt{x})) + C$$ --- Final answers: 1. $$-5 \ln|4 - \sqrt{3} - z| + C$$ or $$5 \ln|z - (4 - \sqrt{3})| + C$$ 2. $$-\frac{1}{4} \left( 2w + 2\sqrt{6} + 5 - (2\sqrt{6} + 5) \ln|2w + 2\sqrt{6} + 5| \right) + C$$ 3. $$2 \sqrt{x} \sin(\sqrt{x}) + 2 \cos(\sqrt{x}) + C$$