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

Consider an integral whose argument can be written as the chain rule derivative of a composite function F(g(x)). To solve the integral, the process involves reversing the chain rule differentiation.Here, a new variable, u, is defined as g(x). Then differentiate u with respect to x. This can be rearranged in terms of du.To change the limits, when x=a, u equals g(a) becomes the new lower limit, and when x=b, u equals g(b) becomes the new upper limit.The original integral is then rewritten by substituting u and du, and replacing the limits a and b with g(a) and g(b), respectively. Integrating the new integrand with respect to u and applying the changing limits gives the final numerical expression.One example of substitution is found in electrical engineering, where it's used to find the total charge that's passed through a circuit over a given time interval. Here total charge is found by calculating the definite integral with respect to time. As the current is given by a complex function, substitution makes the integral easier to solve.

substitution rule applied to definite integrals

When evaluating a definite integral whose integrand matches the structure of a composite function, the substitution method provides an efficient way to simplify the calculation. This method is based on reversing the chain rule from differentiation, allowing a complicated expression to be rewritten in a simpler form. When the integrand contains an inner function and its derivative, substitution naturally reduces the complexity of the problem.The core idea of substitution for definite integrals is summarized by the relationship shown below.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19987/19987_Equation_1_editor_19987.svg" alt="Integration by substitution formula, equation diagram, educational use, calculus concept." data-altupdatedat="1775223722000" data-alt="19987_Equation_1.svg" /></figure>The process begins by identifying the inner function and introducing a new variable to represent it. Differentiating this new variable provides a corresponding differential that replaces the original one. At the same time, the limits of integration are updated by evaluating the inner function at the original endpoints. This ensures that the entire integral is expressed consistently in terms of the new variable, eliminating the need to return to the original variable after integration.Once the substitution is complete, the integral becomes simpler to evaluate because the integrand now depends on a single variable. Applying the updated limits directly yields the final numerical value. This approach streamlines calculations that would otherwise be difficult or impractical using standard integration techniques.Substitution plays a crucial role in various applied fields, including electrical engineering. For example, determining the total electric charge that flows through a circuit over a given time interval requires accumulating the current over that time interval. When the current is described by a complex expression, substitution allows the integral to be evaluated efficiently, making it a practical and powerful tool for analyzing real-world systems.

Substitution Rule Applied to Definite Integrals

When evaluating a definite integral whose integrand matches the structure of a composite function, the substitution method provides an efficient way to ...

substitution-rule-applied-to-definite-integrals

Education

JoVE Core

Calculus

Unit

Integrals

KEY TERMS AND CONCEPTS

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 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.

Using Substitution to Evaluate Definite Integrals | JoVE K12

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calculus

Using Substitution to Evaluate Definite Integrals

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