Evaluating The Definite Integral Of (x - √(x² - 2))⁴⁰⁴⁹

The definite integral $\int_{\sqrt{2}}^{\infty} (x - \sqrt{x^2 - 2})^{4049} dx$ presents a unique challenge that can be tackled using a clever substitution. The integrand, $(x - \sqrt{x^2 - 2})^{4049}$, involves a term with a square root, which often suggests trigonometric or hyperbolic substitutions to simplify the expression. By carefully selecting a substitution, we can transform this integral into a more manageable form, ultimately leading to its evaluation. In this article, we will delve into the step-by-step process of evaluating this integral, shedding light on the techniques and insights required to solve such problems. Understanding how to approach these kinds of integrals is crucial for mastering integral calculus and its applications in various fields. This exploration will not only provide a solution but also enhance your problem-solving skills in calculus. The initial form of the integrand might seem daunting, but with the right approach, it unravels into a solvable problem, showcasing the power and elegance of calculus.

The Strategic Substitution

To simplify the integral, the core idea revolves around making a suitable substitution. Recognizing the structure of the term inside the square root, $x^2 - 2$, a hyperbolic substitution proves to be highly effective. By letting $x = \sqrt{2} \cosh(t)$, we can leverage the identity $\cosh^2(t) - 1 = \sinh^2(t)$ to eliminate the square root. This substitution transforms the integral into a form that is much easier to handle. The differential $dx$ can be found by differentiating $x$ with respect to $t$, giving us $dx = \sqrt{2} \sinh(t) dt$. Now, we must also consider the limits of integration. When $x = \sqrt{2}$, we have $\sqrt{2} = \sqrt{2} \cosh(t)$, implying $\cosh(t) = 1$, which means $t = 0$. As $x$ approaches infinity, $t$ also approaches infinity. So, the new limits of integration are from 0 to infinity. This substitution not only simplifies the algebraic expression but also sets the stage for further simplification using properties of hyperbolic functions and their integrals. The choice of hyperbolic substitution here is not arbitrary; it's a strategic decision based on the form of the integrand, aiming to convert the integral into a simpler, more solvable form. This step is critical in unraveling the complexity of the original integral.

Transforming the Integrand

With the substitution $x = \sqrt{2} \cosh(t)$, the term $x - \sqrt{x^2 - 2}$ becomes $\sqrt{2} \cosh(t) - \sqrt{2 \cosh^2(t) - 2}$. Factoring out the 2 inside the square root gives us $\sqrt{2} \cosh(t) - \sqrt{2(\cosh^2(t) - 1)}$. Using the identity $\cosh^2(t) - 1 = \sinh^2(t)$, the expression simplifies to $\sqrt{2} \cosh(t) - \sqrt{2} \sinh(t)$. We can factor out $\sqrt{2}$ to get $\sqrt{2}(\cosh(t) - \sinh(t))$. Recall that $\cosh(t) = \frac{e^t + e^{-t}}{2}$ and $\sinh(t) = \frac{e^t - e^{-t}}{2}$. Substituting these into the expression, we have $\sqrt{2}(\frac{e^t + e^{-t}}{2} - \frac{e^t - e^{-t}}{2})$, which simplifies to $\sqrt{2}e^{-t}$. Consequently, the original integrand $(x - \sqrt{x^2 - 2})^{4049}$ transforms into $(\sqrt{2}e^{-t})^{4049} = 2^{\frac{4049}{2}}e^{-4049t}$. This transformation is a crucial step in making the integral tractable. By expressing the integrand in terms of exponential functions, we are better positioned to apply standard integration techniques. The strategic use of hyperbolic identities and exponential representations plays a pivotal role in this simplification process, demonstrating the power of mathematical manipulation in solving complex problems.

Setting Up the Transformed Integral

Now that we have transformed the integrand and the limits of integration, we can rewrite the integral. Recall that $dx = \sqrt{2} \sinh(t) dt$ and the new limits are from 0 to infinity. The integral becomes:

$$\int_{0}^{\infty} (\sqrt{2}e^{-t})^{4049} \sqrt{2} \sinh(t) dt$$

We can express $\sinh(t)$ as $\frac{e^t - e^{-t}}{2}$, so the integral is:

$$\int_{0}^{\infty} 2^{\frac{4049}{2}} e^{-4049t} \sqrt{2} \frac{e^t - e^{-t}}{2} dt$$

Simplifying the constants, we get:

$$2^{\frac{4049}{2}} \sqrt{2} \frac{1}{2} \int_{0}^{\infty} e^{-4049t} (e^t - e^{-t}) dt$$

Which simplifies further to:

$$2^{\frac{4048}{2}} \int_{0}^{\infty} (e^{-4048t} - e^{-4050t}) dt$$

This step is crucial as it sets up the integral in a form that is directly integrable. The manipulation of constants and the distribution of the exponential terms allow us to apply standard integration rules. The transformation process has effectively converted a complex integral into a simple difference of two exponential integrals, making it significantly easier to solve. The careful attention to detail in this step ensures that the integral is correctly set up for the final evaluation.

Evaluating the Simplified Integral

With the integral simplified to $2^{2024} \int_{0}^{\infty} (e^{-4048t} - e^{-4050t}) dt$, we can now evaluate it term by term. The integral of $e^{-kt}$ from 0 to infinity is $\frac{1}{k}$. Therefore, the integral becomes:

$$2^{2024} \left[ \int_{0}^{\infty} e^{-4048t} dt - \int_{0}^{\infty} e^{-4050t} dt \right]$$

Evaluating each integral, we get:

$$2^{2024} \left[ \frac{1}{4048} - \frac{1}{4050} \right]$$

We can simplify the fraction inside the brackets by finding a common denominator:

$$2^{2024} \left[ \frac{4050 - 4048}{4048 \cdot 4050} \right] = 2^{2024} \left[ \frac{2}{4048 \cdot 4050} \right]$$

Now, we simplify the expression:

$$2^{2024} \left[ \frac{1}{2024 \cdot 4050} \right] = 2^{2024} \left[ \frac{1}{2024 \cdot 2 \cdot 2025} \right]$$

Further simplification gives us:

$$2^{2023} \left[ \frac{1}{2024 \cdot 2025} \right] = 2^{2023} \left[ \frac{1}{2024} - \frac{1}{2025} \right]$$

This final result matches the expression we aimed to derive. This step demonstrates the straightforward application of integration rules and arithmetic simplification. The ability to break down the integral into manageable parts and apply standard techniques is crucial for successful evaluation. The final simplification showcases the elegance of the solution and validates the initial substitution strategy.

Conclusion

The integral $\int_{\sqrt{2}}^{\infty} (x - \sqrt{x^2 - 2})^{4049} dx$ can be evaluated using a hyperbolic substitution, which simplifies the integrand and allows for straightforward integration. The result is:

$$\int_{\sqrt{2}}^{\infty} (x - \sqrt{x^2 - 2})^{4049} dx = 2^{2023} \left( \frac{1}{2024} - \frac{1}{2025} \right)$$

This problem highlights the power of strategic substitutions in simplifying complex integrals. By recognizing the structure of the integrand and choosing an appropriate substitution, we transformed the integral into a manageable form. The hyperbolic substitution, combined with the properties of hyperbolic functions and exponential representations, proved to be a powerful tool in solving this problem. The step-by-step evaluation demonstrates the importance of careful manipulation and simplification in calculus. The successful evaluation of this integral reinforces the understanding of integral calculus techniques and their application in problem-solving. This comprehensive approach not only provides the solution but also enhances the problem-solving skills necessary for tackling advanced calculus problems. Mastering these techniques is essential for anyone delving into the realms of mathematics, physics, engineering, and other quantitative disciplines.