Name: splitting rational integrals into simpler parts
Uploaded: 2026-01-12T12:24:14.000Z
Duration: 1 min 29 s
Channel: JoVE

유리 함수는 한 다항식과 다른 다항식의 비율로 정의된다. 일반적으로 적절한 유형과 부적절한 유형으로 분류됩니다.적절한 유리 함수의 경우, 분자의 차수는 분모의 차수보다 작다. 여기서 부분분수 분해는 함수를 단순화합니다.다음으로, 양변에 분모를 곱해 분수를 제거합니다. 그 다음 x 에 적절한 값을 부여하면 미지수의 값이 나옵니다.양측을 적분하면 로그항과 적분 상수로 해를 얻을 수 있습니다.부적절한 유리 함수, 즉 분자의 차수가 분모보다 크거나 같은 경우에는 다항 장나눗셈이 먼저 적용된다.이 과정은 함수를 다항식 항과 적절한 분수 두 부분으로 나눕니다.각 부분은 별도로 적분되며, 다항식은 멱법칙으로 풀고, 적절한 분수는 직접 단순화됩니다.다항식 적분은 대수식을 나타내며, 적절한 분수는 로그 항과 상수를 더한 것을 만듭니다.편분이나 나눗셈을 사용하는 유리 함수의 적분은 복잡한 함수를 더 단순한 적분의 합으로 환원합니다.

부분분수 분해를 이용하는 유리함수의 적분

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integration-of-rational-functions-using-partial-fractions

Education

교육

JoVE Core

Calculus

미적분학

Unit

단위

Techniques of Integration

적분 기법

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Integration of Rational Functions Using Partial Fractions

부분분수를 이용한 유리함수 적분

splitting rational integrals into simpler parts

Rational functions are ratios of two polynomials, and their integrals can be simplified by breaking them into smaller parts. A rational function is proper when the degree of the numerator is less than the degree of the denominator. When that is true, partial fraction decomposition rewrites the expression as a sum of simpler rational terms.

To decompose a proper rational function, unknown constants are placed into the simpler terms. The denominator is then multiplied through to remove the fractions. The constants are found by substitution or by matching coefficients. After that, each term can be integrated directly, and the result usually includes logarithmic expressions plus a constant of integration.

An improper rational function has a numerator degree that is greater than or equal to the denominator degree. Before integrating, polynomial long division is used to split the expression into a polynomial part and a proper rational fraction. The polynomial part is integrated with the power rule, which gives algebraic expressions. The remaining proper fraction is then handled with partial fraction decomposition, which again leads to logarithmic terms.

Splitting Rational Integrals Into Simpler Parts

splitting-rational-integrals-into-simpler-parts

integration by parts: indefinite integrals

integration-by-parts-indefinite-integrals

Integration by Parts in AC Circuit Analysis

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Using Parts and Substitution for Area

Integration by parts can make a definite integral easier to solve when the integrand is a product over a fixed interval. The method rewrites the problem as a product at the endpoints minus another definite integral that may be simpler to evaluate. A useful example is the definite integral of inverse tangent, or arctan x. Since there is no direct integration formula for arctan x, the integrand is written as arctan x times the constant function 1. The inverse tangent is chosen to differentiate,...

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integrals-of-powers-of-secant-and-tangent

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integration of rational functions using partial fractions

Rational functions are ratios of two polynomials, and their integrals can be simplified by breaking them into smaller parts. A rational function is proper when the degree of the numerator is less than the degree of the denominator. When that is true, partial fraction decomposition rewrites the expression as a sum of simpler rational terms. To decompose a proper rational function, unknown constants are placed into the simpler terms. The denominator is then multiplied through to remove the...

rationalizing substitutions

Integrating Radicals with Rational Substitution

Integrating radicals can be easier when a rationalizing substitution turns the expression into a rational form. This method is useful when standard techniques do not work well because the integrand contains a radical, such as a cube root. The transcript uses a rod as a real-world example. Its linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Finding the total mass means integrating that density over the length of...

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Riemann sums estimate the area under a curve when an exact definite integral is hard to find. This often happens when a function has no known antiderivative or cannot be written in a closed form. It also happens when the function comes from measured data instead of a formula. The method works by splitting the interval of integration into equal parts. Each subinterval gets a rectangle, and the rectangle height comes from the function value at one chosen point in that part. Adding the rectangle...

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Estimating distance from velocity data is a common use of the trapezoidal rule. In physics and engineering, velocity is often recorded at separate time points instead of as one continuous equation. Numerical integration gives a way to approximate the total displacement from those measurements. The method divides the full time interval into equal parts. For each part, the velocities at the two endpoints are joined with a straight line on a velocity-time graph. That line creates a trapezoid.

simpson's rule i

Estimating Area with Simpson’s Rule

Simpson’s Rule is a numerical integration method for estimating the value of a definite integral. It is useful when an exact antiderivative is hard or impossible to find. The method works by fitting a unique parabola through three equally spaced points on a curve and then integrating that quadratic over the interval. This approach can give a very accurate area estimate from only a few sampled values. It works well for many smoothly varying functions. That makes it a practical tool when a curve...

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Simpson's Rule helps estimate the area under a smooth curve, such as the curved profile of metal roofing sheets. In warehouse roofing, corrugated or curved sheets are used because they add strength, improve water drainage, and support better ventilation. To plan materials and design more accurately, engineers often need the area of these curved surfaces. The sheet shape can often be treated like a parabolic curve because it changes smoothly along its length. That makes numerical integration a...

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Improper Integrals and Exponential Decay

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improper integrals: discontinuous integrands

improper-integrals-discontinuous-integrands

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key terms and concepts

Learn how rational functions are separated into polynomial and fraction parts, then integrated with long division and partial fractions to produce logarithmic results.

Splitting Rational Integrals Into Simpler Parts | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Splitting Rational Integrals Into Simpler Parts

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