indefinite integral

简明释义

不定积分

英英释义

例句

1.When solving differential equations, we often use the indefinite integral 不定积分 to find the general solution.

在解微分方程时，我们常常使用不定积分来找到一般解。

2.The indefinite integral 不定积分 of x^2 is (1/3)x^3 + C.

x^2的不定积分是(1/3)x^3 + C。

3.In calculus, the indefinite integral 不定积分 represents a family of functions.

在微积分中，不定积分表示一族函数。

4.The indefinite integral 不定积分 can be used to determine the original function from its derivative.

可以使用不定积分来确定其导数的原始函数。

5.To find the area under the curve, we need to calculate the indefinite integral 不定积分 of the function.

为了找到曲线下方的面积，我们需要计算该函数的不定积分。

作文

In the field of mathematics, particularly in calculus, the concept of the indefinite integral plays a crucial role. An indefinite integral represents a family of functions whose derivative is the given function. It is essentially the reverse process of differentiation. Understanding this concept is fundamental for students and professionals alike, as it lays the groundwork for more advanced topics such as differential equations and real analysis.To grasp the idea of the indefinite integral, we first need to understand what integration means. Integration is the process of finding the accumulated area under a curve defined by a function. While definite integrals give us a specific numerical value representing this area over a certain interval, indefinite integrals provide a general form without specific limits. This leads us to the notation used in calculus: the symbol ∫ is employed to denote integrals, and when we write ∫f(x)dx, we are looking for a function F(x) such that F'(x) = f(x).For example, if we take the function f(x) = 2x, the indefinite integral would be ∫2xdx = x² + C, where C is the constant of integration. This constant is crucial because the process of differentiation removes any constant term, making it impossible to determine the exact value of C without additional information. Therefore, the indefinite integral gives us a whole family of functions that differ only by a constant.The importance of the indefinite integral extends beyond pure mathematics. In physics, for instance, it is used to calculate quantities such as displacement, area, and volume. When working with motion, the relationship between velocity and position can be understood through the indefinite integral. If we know the velocity function of an object, integrating this function will yield the position function, which is essential for understanding the motion of objects.Moreover, the indefinite integral is also instrumental in solving problems related to growth and decay. In biology, for example, the growth of populations can often be modeled using differential equations, which require integration for their solutions. The indefinite integral allows us to find the general solution to these equations, giving us insights into how populations change over time.In conclusion, the indefinite integral is a fundamental concept in calculus that has far-reaching implications across various fields of study. Its ability to provide a general form of functions based on their derivatives makes it an invaluable tool for mathematicians, scientists, and engineers. By mastering the indefinite integral, one gains the skills necessary to tackle complex problems and deepen their understanding of the mathematical world around them. As we continue to explore the intricacies of calculus, the indefinite integral will undoubtedly remain a cornerstone of our mathematical toolkit, enabling us to unlock the secrets of change and accumulation in both theoretical and practical applications.

在数学领域，特别是在微积分中，不定积分的概念起着至关重要的作用。不定积分表示一组函数，其导数是给定的函数。它本质上是微分的反过程。理解这一概念对于学生和专业人士来说都是基础，因为它为更高级的主题如微分方程和实分析奠定了基础。要掌握不定积分的概念，我们首先需要理解积分的含义。积分是指寻找由函数定义的曲线下的累积面积的过程。当定积分给我们一个特定的数值，表示在某个区间内的面积时，不定积分则提供了没有特定限制的一般形式。这引导我们使用微积分中的符号：符号∫用于表示积分，当我们写∫f(x)dx时，我们在寻找一个函数F(x)，使得F'(x) = f(x)。例如，如果我们取函数f(x) = 2x，则不定积分将是∫2xdx = x² + C，其中C是积分常数。这个常数非常关键，因为微分过程去除了任何常数项，使得没有额外信息就无法确定C的确切值。因此，不定积分给我们提供了一整组仅因常数而异的函数。不定积分的重要性超越了纯数学。在物理学中，例如，它用于计算位移、面积和体积等数量。当处理运动时，速度与位置之间的关系可以通过不定积分来理解。如果我们知道一个物体的速度函数，积分这个函数将产生位置函数，这对于理解物体的运动至关重要。此外，不定积分在解决与增长和衰减相关的问题中也很重要。在生物学中，例如，种群的增长通常可以使用微分方程建模，而这些方程的解需要积分。不定积分使我们能够找到这些方程的一般解，从而深入了解种群随时间变化的情况。总之，不定积分是微积分中的一个基本概念，在各个研究领域都有深远的影响。它基于导数提供函数的一般形式，使其成为数学家、科学家和工程师的宝贵工具。通过掌握不定积分，人们获得了解决复杂问题所需的技能，并加深了对周围数学世界的理解。随着我们继续探索微积分的复杂性，不定积分无疑将继续作为我们数学工具箱的基石，使我们能够揭示理论和实际应用中变化和累积的奥秘。