Evaluate The Indefinite Integral ∫ (sin(1/x^4) / X^5) Dx. Include The Constant Of Integration, C.

Jun 21, 2025 by ADMIN 98 views

Evaluating the Indefinite Integral of (sin(1/x^4) / x^5) dx

In this article, we will delve into the process of evaluating the indefinite integral of the function f(x) = sin(1/x^4) / x^5. This type of integral often requires a clever application of substitution techniques, a fundamental tool in integral calculus. We'll break down the steps involved, providing a clear and comprehensive explanation to help you grasp the underlying concepts. Understanding these methods is crucial for tackling more complex integrals and is a cornerstone of advanced calculus.

Understanding Indefinite Integrals

Before we dive into the specifics of this integral, let's briefly review the concept of indefinite integrals. An indefinite integral, denoted by the integral symbol ∫, represents the family of functions whose derivative is the integrand (the function inside the integral). In simpler terms, we are looking for a function whose derivative is equal to sin(1/x^4) / x^5. Since the derivative of a constant is zero, we always add a constant of integration, usually denoted by C, to the result of an indefinite integral. This constant represents the ambiguity in the antiderivative, as there are infinitely many functions that differ only by a constant and have the same derivative.

The process of finding an indefinite integral is essentially the reverse of differentiation. We need to identify a function that, when differentiated, yields the integrand. This can sometimes be straightforward, but more often than not, it requires employing various techniques such as substitution, integration by parts, or trigonometric identities. The key is to recognize patterns and choose the appropriate method to simplify the integral.

For our specific problem, ∫ (sin(1/x^4) / x^5) dx, the presence of a composite function (sin of something) and a related term in the denominator (x^5) strongly suggests that the method of u-substitution will be particularly effective. This technique allows us to simplify the integral by replacing a complex expression with a single variable, making the integration process more manageable. Let's explore how this method can be applied to solve our integral.

Applying U-Substitution

U-substitution, also known as substitution, is a powerful technique used to simplify integrals by changing the variable of integration. The core idea is to identify a suitable expression within the integrand, call it u, and replace it along with its differential du. This often transforms the integral into a simpler form that can be readily integrated using standard integration rules.

In our case, the integrand is sin(1/x^4) / x^5. Observing the structure of the function, we can see that the expression 1/x^4 appears inside the sine function. This suggests that we should choose u = 1/x^4. This choice is strategic because the derivative of 1/x^4 will involve a term with x^5 in the denominator, which is already present in our integrand. This alignment is crucial for the success of the u-substitution method.

Let's proceed with the substitution. If u = 1/x^4, we can rewrite this as u = x^(-4). Now, we need to find the differential du. Differentiating u with respect to x, we get:

du/dx = -4x^(-5)

Multiplying both sides by dx, we obtain:

du = -4x^(-5) dx

We can rewrite this as:

du = -4(1/x^5) dx

Notice that we have the term (1/x^5) dx in our original integral. We can isolate this term by dividing both sides of the equation by -4:

(-1/4) du = (1/x^5) dx

Now we have everything we need to perform the substitution. We replace 1/x^4 with u and (1/x^5) dx with (-1/4) du in the original integral:

∫ (sin(1/x^4) / x^5) dx = ∫ sin(u) * (-1/4) du

Integrating with the New Variable

Now that we have performed the u-substitution, our integral has transformed into a much simpler form: ∫ sin(u) * (-1/4) du. This integral is significantly easier to evaluate than the original one. We can pull the constant factor (-1/4) out of the integral:

∫ sin(u) * (-1/4) du = (-1/4) ∫ sin(u) du

We now need to find the indefinite integral of sin(u) with respect to u. Recall that the derivative of cos(u) is -sin(u). Therefore, the indefinite integral of sin(u) is -cos(u). So, we have:

(-1/4) ∫ sin(u) du = (-1/4) * (-cos(u)) + C

Simplifying, we get:

(1/4) cos(u) + C

Remember that C represents the constant of integration, which is always added to indefinite integrals.

At this point, we have successfully integrated with respect to the variable u. However, our original problem was in terms of x, so we need to substitute back to express the result in terms of x. This is the final step in the u-substitution process.

Substituting Back to the Original Variable

The final step in the u-substitution process is to substitute back the original variable, x, into our result. We defined u as 1/x^4, so we replace u with 1/x^4 in our expression:

(1/4) cos(u) + C = (1/4) cos(1/x^4) + C

Therefore, the indefinite integral of sin(1/x^4) / x^5 with respect to x is (1/4) cos(1/x^4) + C. This is our final answer.

We have successfully evaluated the indefinite integral by using the u-substitution method. This technique allowed us to transform a complex integral into a simpler one, which we could then integrate using basic integration rules. The key to u-substitution is to choose a suitable substitution that simplifies the integrand, making it easier to integrate.

Conclusion

In conclusion, we have successfully evaluated the indefinite integral ∫ (sin(1/x^4) / x^5) dx using the method of u-substitution. The solution is (1/4) cos(1/x^4) + C, where C is the constant of integration. This example demonstrates the power and versatility of u-substitution as a technique for simplifying and solving integrals. Mastering this technique is essential for success in integral calculus and related fields.

Key takeaways from this process include:

Recognizing when u-substitution is appropriate: Look for composite functions and terms that are derivatives of expressions within those functions.
Choosing the right substitution: Selecting the appropriate u is crucial for simplifying the integral.
Finding the differential du: Accurately calculating du in terms of dx is essential for the substitution.
Substituting and integrating: Replacing the original variables and integrating with respect to u.
Substituting back: Expressing the final result in terms of the original variable.

By following these steps, you can effectively apply u-substitution to solve a wide range of indefinite integrals. The ability to skillfully manipulate and solve integrals is a fundamental skill in mathematics and has broad applications in physics, engineering, and other scientific disciplines. Remember to always add the constant of integration, C, when evaluating indefinite integrals, as it represents the family of functions that have the same derivative.

This detailed walkthrough provides a solid foundation for understanding and applying u-substitution. Practice with various examples will further solidify your understanding and improve your problem-solving skills in integral calculus.