1. **Stating the problem:** We need to solve three integrals: - $$\int 4x e^x \, dx$$ (using integration by parts) - $$\int \ln(x) \, dx$$ (using integration by parts) - $$\int \frac{4x}{x^2 - 3} \, dx$$ (using substitution method) 2. **Integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$ This formula is used when the integral is a product of two functions. 3. **First integral: $$\int 4x e^x \, dx$$** - Let $$u = 4x$$ so $$du = 4 dx$$ - Let $$dv = e^x dx$$ so $$v = e^x$$ - Applying integration by parts: $$\int 4x e^x dx = 4x e^x - \int 4 e^x dx = 4x e^x - 4 e^x + C$$ 4. **Second integral: $$\int \ln(x) \, dx$$** - Let $$u = \ln(x)$$ so $$du = \frac{1}{x} dx$$ - Let $$dv = dx$$ so $$v = x$$ - Applying integration by parts: $$\int \ln(x) dx = x \ln(x) - \int x \cdot \frac{1}{x} dx = x \ln(x) - \int 1 dx = x \ln(x) - x + C$$ 5. **Substitution method for $$\int \frac{4x}{x^2 - 3} dx$$** - Let $$t = x^2 - 3$$ so $$dt = 2x dx$$ - Rewrite the integral: $$\int \frac{4x}{x^2 - 3} dx = \int \frac{4x}{t} dx$$ - Express $$dx$$ in terms of $$dt$$: $$dt = 2x dx \Rightarrow dx = \frac{dt}{2x}$$ - Substitute back: $$\int \frac{4x}{t} dx = \int \frac{4x}{t} \cdot \frac{dt}{2x} = \int \frac{4x}{t} \cdot \frac{dt}{2x} = \int \frac{4}{2t} dt = \int \frac{2}{t} dt$$ - Integrate: $$2 \int \frac{1}{t} dt = 2 \ln|t| + C = 2 \ln|x^2 - 3| + C$$ **Final answers:** - $$\int 4x e^x dx = 4x e^x - 4 e^x + C$$ - $$\int \ln(x) dx = x \ln(x) - x + C$$ - $$\int \frac{4x}{x^2 - 3} dx = 2 \ln|x^2 - 3| + C$$