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Evaluating the Definite Integral: ∫₁² e^(1/x⁴) / x⁵ dx

Evaluating definite integrals is a cornerstone of calculus, crucial for numerous applications in physics, engineering, and other fields. This article delves into the detailed evaluation of the definite integral: ∫₁² e^(1/x⁴) / x⁵ dx. We'll explore various techniques, focusing on substitution and its practical application to solve this specific problem. We'll also touch upon the broader context of definite integrals and their significance.

Understanding Definite Integrals

A definite integral, represented as ∫_a^b f(x) dx, calculates the signed area between the curve of a function f(x) and the x-axis, bounded by the limits of integration 'a' and 'b'. The fundamental theorem of calculus connects definite integrals to antiderivatives, providing a powerful method for their evaluation. The process involves finding an antiderivative (also known as an indefinite integral) of the function, evaluating it at the upper and lower limits of integration, and then finding the difference.

The Problem: ∫₁² e^(1/x⁴) / x⁵ dx

Our target integral presents a unique challenge: ∫₁² e^(1/x⁴) / x⁵ dx. This integral isn't immediately solvable using basic integration rules. However, a clever substitution can dramatically simplify the integrand, making it amenable to straightforward integration.

The Power of Substitution

Substitution, also known as u-substitution, is a powerful integration technique that transforms complex integrals into simpler ones. The core idea is to replace a part of the integrand with a new variable, 'u', and then express the rest of the integrand in terms of 'u' and 'du'. The choice of substitution is often guided by the structure of the integrand, looking for a function and its derivative.

Applying Substitution to Our Integral

Let's carefully choose our substitution for ∫₁² e^(1/x⁴) / x⁵ dx. Observe that the exponent of 'e' involves 1/x⁴, and the denominator contains x⁵. This suggests a strategic substitution:

Let u = 1/x⁴

Now, we need to find du in terms of dx:

du = -4x⁻⁵ dx => du = -4/x⁵ dx

This is excellent! Notice that we have 1/x⁵ dx in our original integral. Let's rewrite our integral using this substitution:

∫₁² e^(1/x⁴) / x⁵ dx = ∫₁² e^u (-1/4) du

Adjusting the Limits of Integration

A crucial step when using substitution in definite integrals is to adjust the limits of integration to reflect the new variable 'u'. Our original limits were x = 1 and x = 2. We need to find the corresponding values of 'u':

• When x = 1, u = 1/1⁴ = 1
• When x = 2, u = 1/2⁴ = 1/16

Therefore, our integral transforms completely to:

∫₁^(1/16) e^u (-1/4) du

Completing the Integration

This integral is now straightforward to evaluate. The antiderivative of e^u is simply e^u. So we have:

(-1/4) ∫₁^(1/16) e^u du = (-1/4) [e^u]₁^(1/16)

Evaluating the Antiderivative at the Limits

Now, we evaluate the antiderivative at the adjusted limits:

(-1/4) [e^(1/16) - e¹] = (-1/4) [e^(1/16) - e]

The Final Result

Therefore, the final evaluation of the definite integral ∫₁² e^(1/x⁴) / x⁵ dx is:

(-1/4) [e^(1/16) - e]

This represents the exact value of the definite integral. You can use a calculator to obtain a numerical approximation if needed.

Significance and Applications of Definite Integrals

Definite integrals have far-reaching applications across various disciplines:

1. Calculating Areas

The most fundamental application is calculating areas under curves. This is directly relevant in geometry, finding areas of irregular shapes.

2. Physics

Definite integrals are crucial in physics for:

• Work: Calculating the work done by a variable force.
• Displacement: Determining the displacement of a particle given its velocity function.
• Center of Mass: Finding the center of mass of a system.
• Moments of Inertia: Calculating the moments of inertia for rotating objects.

3. Engineering

Engineers use definite integrals for:

• Stress and Strain Analysis: Calculating stress and strain in structures.
• Fluid Dynamics: Determining fluid flow rates and pressures.
• Signal Processing: Analyzing and filtering signals.

4. Probability and Statistics

Definite integrals play a pivotal role in probability and statistics for:

• Probability Density Functions: Calculating probabilities using probability density functions.
• Expected Values: Computing expected values of random variables.

Beyond Substitution: Other Integration Techniques

While substitution proved sufficient for our specific problem, other techniques are essential for tackling a wider range of integrals. These include:

• Integration by Parts: Used for integrals involving products of functions.
• Partial Fraction Decomposition: Useful for integrals involving rational functions.
• Trigonometric Substitution: Employed when trigonometric functions appear in the integrand.
• Numerical Integration: Used when analytical solutions are difficult or impossible to obtain. Methods like Simpson's rule and the trapezoidal rule provide numerical approximations.

Conclusion

Evaluating definite integrals is a fundamental skill in calculus with wide-ranging applications. The problem presented, ∫₁² e^(1/x⁴) / x⁵ dx, highlights the effectiveness of substitution as a technique for simplifying complex integrals. Understanding various integration methods and their contexts is key to mastering calculus and successfully solving problems in diverse scientific and engineering domains. Remember to always meticulously check your work and ensure you have correctly applied the rules of integration, particularly when dealing with limits and substitutions. This attention to detail ensures accuracy in your solutions.