91Ó°ÊÓ

The power rule is a fundamental concept in calculus used to evaluate integrals and derivatives. It is especially useful for integrating functions of the form \(x^n\). The power rule for integration states that to integrate \(x^n\), you increase the exponent by one and divide by the new exponent. This can be summarized as follows:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), for \(n eq -1\).

In our exercise, the integrand \(x + \frac{2}{x^2} - \frac{1}{x^4}\) involves applying the power rule to different powers of \(x\). Each term is integrated individually, considering their respective powers:

• The term \(x = x^1\) becomes \(\frac{x^2}{2}\).
• The term \(\frac{2}{x^2} = 2x^{-2}\) becomes \(-\frac{2}{x}\).
• The term \(-\frac{1}{x^4} = -x^{-4}\) becomes \(\frac{1}{3x^3}\).

The power rule simplifies the integration process, turning the task of integrating polynomial terms into a straightforward calculation.

Differentiation is the reverse process of integration, confirming that the integral derived is accurate. In mathematics, it is common practice to check your solution by differentiating it, particularly with indefinite integrals like the one in this exercise.By differentiating the result of our integration \(\left( \frac{x^2}{2} - \frac{2}{x} + \frac{1}{3x^3} + C \right)\), we aim to retrieve the original simplified integrand. This is done by applying the basic rules of differentiation:

• The term \(\frac{x^2}{2}\) differentiates to \(x\).
• The term \(-\frac{2}{x}\) differentiates to \(\frac{2}{x^2}\).
• The term \(\frac{1}{3x^3}\) differentiates to \(-\frac{1}{x^4}\).

Upon combining these results, the expression matches the simplified integrand. This consistency verifies that the integration process was correct.

The verification of integrals is essential in calculus, ensuring the accuracy of evaluated integrals. This process not only confirms the correctness of the integral but also strengthens understanding of the relationship between differentiation and integration. In practical scenarios, this verification acts as a safeguard against errors in computation or conceptual misunderstandings. After integrating, differentiate your result to confirm it matches the original integrand. This practice consolidates learning and provides a deeper grasp of calculus principles. Moreover, seeing differentiation as a means of verification reinforces the inverse relationship between differentiation and integration. This is foundational in calculus:

• Integrating a function involves finding the antiderivative.
• Differentiating the antiderivative should yield the original function.

Thus, verification becomes a powerful tool both for verifying accuracy and for deepening comprehension of integral calculus.