Evaluate The Definite Integral. 2 E 1/x5 X6 Dx 1

Evaluating the Definite Integral: ∫₂¹ e^(1/x⁵) / x⁶ dx

Evaluating definite integrals is a cornerstone of calculus, crucial for numerous applications in physics, engineering, and other scientific fields. This article delves into the process of evaluating a specific definite integral, ∫₂¹ e^(1/x⁵) / x⁶ dx, demonstrating various techniques and providing a step-by-step solution. We'll explore the intricacies of substitution, the importance of proper limits of integration, and the significance of understanding the underlying concepts.

Understanding the Problem: ∫₂¹ e^(1/x⁵) / x⁶ dx

The integral presents a seemingly complex function. However, with the right approach, it simplifies significantly. The key lies in recognizing the potential for substitution to simplify the integrand. Our focus will be on finding a suitable substitution that transforms the integral into a more manageable form.

The Power of Substitution: A Strategic Approach

The core strategy for solving this integral lies in the method of substitution (also known as u-substitution). The goal is to find a suitable substitution that will simplify the integrand, making it easier to integrate. Let's analyze the integrand: e^(1/x⁵) / x⁶. We observe that the derivative of 1/x⁵ is related to the denominator, x⁶.

Let's choose our substitution:

u = 1/x⁵ = x⁻⁵

Now, we need to find the derivative of u with respect to x:

du/dx = -5x⁻⁶ = -5/x⁶

We can rearrange this to solve for dx:

dx = -x⁶/5 du

Now, we substitute u and dx into our original integral:

∫₂¹ e^(1/x⁵) / x⁶ dx = ∫₂¹ eᵘ (-x⁶/5 du) / x⁶

Notice how the x⁶ terms cancel out, significantly simplifying the integral:

∫₂¹ eᵘ (-1/5) du = (-1/5) ∫₂¹ eᵘ du

Integrating the Simplified Expression

The integral is now considerably easier to evaluate. The integral of eᵘ with respect to u is simply eᵘ:

(-1/5) ∫₂¹ eᵘ du = (-1/5) [eᵘ]₂¹

Handling the Limits of Integration

This is a crucial step often missed by beginners. Since we performed a substitution, we must change the limits of integration from x-values to u-values. Recall that u = 1/x⁵:

• When x = 2, u = 1/2⁵ = 1/32
• When x = 1, u = 1/1⁵ = 1

Therefore, our integral becomes:

(-1/5) [eᵘ]₁⁽¹/³²⁾

Evaluating the Definite Integral

Now, substitute the limits of integration into the expression:

(-1/5) [e¹ - e^(¹/³²)] = (-1/5) [e - e^(¹/³²)]

This is our final answer. Although we cannot simplify it further to an exact numerical value without a calculator, we have successfully evaluated the definite integral using the method of substitution.

Numerical Approximation and Significance

While the exact expression (-1/5) [e - e^(¹/³²)] is perfectly valid, it's often helpful to obtain a numerical approximation using a calculator:

(-1/5) [e - e^(¹/³²)] ≈ -0.1264

This numerical value offers a practical interpretation of the area under the curve represented by the integral. The negative sign indicates that the area lies predominantly below the x-axis.

Further Applications and Extensions

The techniques demonstrated here are applicable to a broader range of definite integrals involving exponential functions and rational functions. Understanding substitution is fundamental for mastering more advanced integration techniques like integration by parts and trigonometric substitution. This problem highlights the importance of carefully choosing a suitable substitution to simplify the integrand and accurately handle the limits of integration.

Common Mistakes and Troubleshooting

Several common mistakes can arise when evaluating definite integrals using substitution:

• Forgetting to change the limits of integration: This is a critical error. Always transform the limits of integration from the original variable to the substituted variable.
• Incorrectly computing the derivative of the substitution: A slight error in calculating du/dx will propagate through the entire solution, leading to an inaccurate result. Double-checking this step is crucial.
• Ignoring the constant of integration in definite integrals: While the constant of integration cancels out when evaluating a definite integral, forgetting it during the intermediate steps can confuse the process.
• Misinterpreting the signs: Pay close attention to signs, especially when dealing with negative exponents or derivatives.

Conclusion: Mastering Definite Integrals

This detailed walkthrough illustrates the systematic approach required to evaluate a seemingly complex definite integral. Through the application of substitution and careful attention to detail, we successfully found a solution and gained a better understanding of the numerical significance of the result. Mastering these techniques is paramount for success in calculus and its numerous applications across scientific disciplines. Remember to practice various examples and refine your skills to handle increasingly complex integration problems effectively. The key is a structured approach, careful execution of steps, and a solid understanding of the underlying mathematical principles.