1. **Stating the problem:** Evaluate the integral $$\int_{-\infty}^{\infty} \sqrt{x^2 + \tan(\theta)} \cdot (\cos(\theta) \tan(\theta)) \, dx = \pi + 3$$ and understand the relationship between the integral and the given expression. 2. **Analyzing the integral:** The integral is over all real numbers from $-\infty$ to $\infty$ of the function $$f(x) = \sqrt{x^2 + \tan(\theta)} \cdot (\cos(\theta) \tan(\theta)).$$ 3. **Key observations:** - The term $\sqrt{x^2 + \tan(\theta)}$ is an even function in $x$ because $x^2$ is even. - The factor $\cos(\theta) \tan(\theta)$ is independent of $x$ and can be treated as a constant with respect to integration. 4. **Simplify the integral using even function property:** Since the integrand is even in $x$, $$\int_{-\infty}^{\infty} f(x) \, dx = 2 \int_0^{\infty} f(x) \, dx.$$ 5. **Rewrite the integral:** $$\int_{-\infty}^{\infty} \sqrt{x^2 + \tan(\theta)} \cdot (\cos(\theta) \tan(\theta)) \, dx = 2 \cos(\theta) \tan(\theta) \int_0^{\infty} \sqrt{x^2 + \tan(\theta)} \, dx.$$ 6. **Evaluate the integral $$\int_0^{\infty} \sqrt{x^2 + a} \, dx$$ where $a = \tan(\theta)$:** This integral diverges to infinity for positive $a$ because the integrand behaves like $x$ for large $x$. 7. **Conclusion:** The integral as stated diverges unless there is additional context or constraints on $\theta$ or the integral. The equation $$\int_{-\infty}^{\infty} \sqrt{x^2 + \tan(\theta)} (\cos(\theta) \tan(\theta)) \, dx = \pi + 3$$ suggests a specific value of $\theta$ or a different interpretation. 8. **Summary:** - The integral involves an even function multiplied by a constant. - The integral over $x$ from $-\infty$ to $\infty$ of $\sqrt{x^2 + a}$ diverges. - Therefore, the equation holds only under special conditions or interpretations not provided. **Final answer:** The integral diverges for general $\theta$, so the equation as given cannot hold without additional constraints.