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

Evaluating definite integrals can often seem daunting, especially when they involve complex functions. However, with the right techniques and a systematic approach, even seemingly complicated integrals can be solved. Let's dive into the process of evaluating the definite integral ∫[1 to 2] e^(1/x^5) / x^6 dx.

Understanding the Definite Integral

Before we delve into the specifics of this integral, it's essential to understand what a definite integral represents. A definite integral calculates the area under a curve between two specified limits. In mathematical terms, given a function f(x), the definite integral from a to b is written as:

∫[a to b] f(x) dx

Here, a and b are the limits of integration, and the result is a numerical value representing the signed area between the curve f(x) and the x-axis from x = a to x = b.

Breaking Down the Integral ∫[1 to 2] e^(1/x^5) / x^6 dx

Now, let's focus on the given integral:

∫[1 to 2] e^(1/x^5) / x^6 dx

This integral might appear intimidating at first glance, but with a strategic approach, we can simplify it. The key to solving this integral lies in recognizing a suitable substitution.

Step 1: Identifying a Suitable Substitution

The function e^(1/x^5) and the term 1/x^6 suggest that a u-substitution might be effective. We look for a function and its derivative within the integral. Let's consider:

u = 1/x^5

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

du/dx = -5/x^6

Rearranging this, we get:

dx = -x^6/5 du

Step 2: Rewriting the Integral in Terms of u

We now substitute u and dx into the original integral:

∫ e^u / x^6 (-x^6/5) du

Notice that the x^6 terms cancel out, simplifying the integral significantly:

∫ e^u (-1/5) du

We can pull the constant -1/5 out of the integral:

(-1/5) ∫ e^u du

Step 3: Changing the Limits of Integration

Since we are dealing with a definite integral, we need to change the limits of integration from x values to u values.

When x = 1: u = 1/(1^5) = 1

When x = 2: u = 1/(2^5) = 1/32

So, our new limits of integration are from 1 to 1/32. The integral now becomes:

(-1/5) ∫[1 to 1/32] e^u du

Step 4: Evaluating the Integral

The integral of e^u with respect to u is simply e^u. Therefore, we have:

(-1/5) [e^u] from 1 to 1/32

Now, we evaluate the antiderivative at the upper and lower limits:

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

Step 5: Simplifying the Result

We can distribute the -1/5:

(-1/5)e^(1/32) + (1/5)e

This is the exact value of the definite integral. To get an approximate numerical value, we can use a calculator:

(-1/5)e^(1/32) + (1/5)e ≈ (-1/5)(1.0317) + (1/5)(2.7183) ≈ -0.20634 + 0.54366 ≈ 0.33732

Thus, the approximate value of the definite integral ∫[1 to 2] e^(1/x^5) / x^6 dx is approximately 0.33732.

Common Mistakes to Avoid

When evaluating definite integrals, several common mistakes can occur. Being aware of these pitfalls can help ensure accuracy:

• Forgetting to Change the Limits of Integration: When using u-substitution for definite integrals, it's crucial to change the limits of integration to the new variable u. Failing to do so will require converting back to the original variable after finding the antiderivative, which can be more complex.
• Incorrectly Finding the Derivative: A mistake in finding the derivative of the substitution variable can lead to an incorrect integral. Always double-check the derivative before proceeding.
• Algebraic Errors: Simple algebraic errors, such as incorrect simplification or distribution, can significantly impact the final result.
• Forgetting the Constant of Integration for Indefinite Integrals: While the constant of integration C is not needed for definite integrals, it is essential for indefinite integrals.
• Misapplying Integration Techniques: Choosing the wrong integration technique, such as attempting integration by parts when a simple substitution would suffice, can complicate the problem unnecessarily.

Alternative Methods

While u-substitution is the most straightforward approach for this particular integral, it's worth noting that other methods might be applicable in different scenarios. These include:

• Integration by Parts: Useful for integrals involving products of functions.
• Trigonometric Substitution: Effective for integrals containing square roots of quadratic expressions.
• Partial Fraction Decomposition: Used for integrating rational functions.
• Numerical Integration: Techniques like Simpson's rule or the trapezoidal rule can be used to approximate definite integrals when an analytical solution is difficult or impossible to find.

Practical Applications of Definite Integrals

Definite integrals are not just abstract mathematical concepts; they have numerous practical applications in various fields:

• Physics: Calculating displacement, velocity, and acceleration; determining the work done by a force; finding the center of mass and moment of inertia.
• Engineering: Analyzing circuits, designing structures, and modeling fluid flow.
• Economics: Calculating consumer and producer surplus, determining present and future values of investments.
• Statistics: Finding probabilities and expected values in probability distributions.
• Computer Graphics: Rendering images and creating realistic simulations.

Advanced Techniques and Considerations

As you progress in calculus, you'll encounter more complex integrals that require advanced techniques. Some of these include:

• Improper Integrals: Integrals with infinite limits of integration or integrals where the function has a singularity within the interval of integration.
• Contour Integration: A technique used in complex analysis to evaluate integrals along paths in the complex plane.
• Special Functions: Integrals involving special functions like the gamma function, beta function, and error function.

Importance of Practice

Mastering the evaluation of definite integrals requires consistent practice. Work through a variety of problems, starting with simpler ones and gradually increasing in complexity. Pay attention to the details, and don't be afraid to seek help when needed. The more you practice, the more comfortable and confident you'll become in your ability to solve even the most challenging integrals.

Step-by-Step Summary

To recap, here’s a step-by-step summary of how we evaluated the definite integral ∫[1 to 2] e^(1/x^5) / x^6 dx:

1. Identify a Suitable Substitution: Recognize that u = 1/x^5 is a good choice because its derivative is related to 1/x^6.
2. Find the Derivative: Calculate du/dx = -5/x^6 and rearrange to find dx = -x^6/5 du.
3. Rewrite the Integral in Terms of u: Substitute u and dx into the integral to get (-1/5) ∫ e^u du.
4. Change the Limits of Integration: Convert the limits from x values to u values: x = 1 becomes u = 1, and x = 2 becomes u = 1/32.
5. Evaluate the Integral: Integrate (-1/5) ∫[1 to 1/32] e^u du to get (-1/5) [e^u] from 1 to 1/32.
6. Simplify the Result: Evaluate the antiderivative at the limits and simplify to obtain (-1/5)e^(1/32) + (1/5)e ≈ 0.33732.

Conclusion

Evaluating definite integrals, such as ∫[1 to 2] e^(1/x^5) / x^6 dx, involves a systematic approach that combines strategic substitution with careful attention to detail. By mastering techniques like u-substitution and understanding common pitfalls, you can confidently tackle a wide range of integration problems. Remember to practice consistently and apply these concepts to real-world applications to solidify your understanding. The journey through calculus is challenging but rewarding, and each integral you solve brings you one step closer to mastering this powerful tool.