Name: using substitution to evaluate definite integrals
Uploaded: 2026-01-12T12:24:59.000Z
Duration: 1 min 24 s
Channel: JoVE

考虑一个积分，其被积函数可以表示为复合函数 F(g(x)) 的链式法则导数。求解该积分的过程涉及对链式法则求导进行逆向运算。此处定义了一个新变量 u，即 g(x)。然后对 u 关于 x 求导。这可以重新表示为 du 的形式。当 x=a 时，u 等于 g(a) 变为新的下限；当 x=b 时，u 等于 g(b) 变为新的上限，以此来改变积分限。然后通过代入 u 和 du 重写原始积分，并将积分限 a 和 b 分别替换为 g(a) 和 g(b)。对新的被积函数关于 u 进行积分，并应用变化后的积分限，即可得到最终的数值表达式。一个代换法的应用实例出现在电气工程中，该方法用于求解在给定时间间隔内通过电路的总电荷量。此处通过计算关于时间的定积分来求得总电荷。由于电流由一个复杂函数表示，使用代换法可使积分更易于求解。

定积分的换元法

Substitution Rule Applied to Definite Integrals

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substitution-rule-applied-to-definite-integrals

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Integrals

积分

KEY TERMS AND CONCEPTS

关键术语与概念

using substitution to evaluate definite integrals

Substitution makes definite integrals easier to evaluate when the integrand is a composite function. It works by reversing the chain rule from differentiation. When the expression includes an inner function and its derivative, substitution can simplify the setup quickly.

The method starts by choosing the inner function and naming it with a new variable. Then its differential is found and used to replace the original variable terms. The limits of integration are also changed by plugging the original endpoints into the inner function. This keeps the integral written in one variable from start to finish.

After the change of variables, the integral usually becomes simpler to solve. Because the updated limits are used directly, there is no need to convert back to the original variable at the end. This makes the final numerical answer easier to obtain.

Substitution is useful when standard integration would be difficult or slow. It is also important in applied science and engineering. For example, in electrical engineering, it can help find the total electric charge that flows through a circuit over a time interval by integrating the current. If the current is given by a complicated expression, substitution makes the calculation more practical.

Using Substitution to Evaluate Definite Integrals

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substitution rule applied to definite integrals

Substitution makes definite integrals easier to evaluate when the integrand is a composite function. It works by reversing the chain rule from differentiation. When the expression includes an inner function and its derivative, substitution can simplify the setup quickly. The method starts by choosing the inner function and naming it with a new variable. Then its differential is found and used to replace the original variable terms. The limits of integration are also changed by plugging the...

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key terms and concepts

Learn how substitution simplifies definite integrals by changing variables, updating limits, and solving the integral directly. Includes a circuit current example.

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calculus

Using Substitution to Evaluate Definite Integrals

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