Name: solving secant and tangent integrals
Uploaded: 2026-01-12T12:24:14.000Z
Duration: 1 min 18 s
Channel: JoVE

Integrale, die Tangential- und Sekantenpotenzen enthalten, können mittels Substitution gelöst werden, und der Ansatz hängt davon ab, ob die Potenz der Sekante gerade oder die Tangentialpotenz ungerade ist.Wenn die Potenz der Sekant gerade ist, wähle Tangente als neue Variable, da ihre Ableitung das Sekantenquadrat ist, das herausfaktorisiert werden kann.Die verbleibende gerade Potenz der Sekante kann dann vollständig in Tangentialterme umgewandelt werden, indem die Identität verwendet wird, die das Sekantenquadrat mit dem Tangentialquadrat verbindet.Diese Substitution verwandelt das Integral in ein Polynom, das dann integriert und wieder in die ursprüngliche Variable umgewandelt wird.Ist die Tangentenpotenz ungerade, wird eine ähnliche Methode verwendet. Sei Sekant die neue Variable, da ihre Ableitung Sekante mal Tangente herausfaktorisiert werden kann.Verwenden Sie eine trigonometrische Identität, um die verbleibende gerade Potenz der Tangente in Bezug auf Sekante umzuschreiben.Nach der Substitution wird das Integral zu einem Polynomausdruck, der direkt integriert werden kann.Schließlich wird das Ergebnis wieder in die ursprüngliche trigonometrische Funktion umgewandelt, um die Lösung zu vervollständigen.

integrale von potenzen des tangens und der sekans

Integrale, die Potenzen des Tangens und der Sekans enthalten, werden &#xFC;blicherweise durch Substitution berechnet, wobei die Vorgehensweise von der Parit&#xE4;t der Exponenten abh&#xE4;ngt. Das Verfahren basiert darauf, einen Teil des Integranden mit der Ableitung einer geeigneten trigonometrischen Funktion zu kombinieren und die verbleibenden Faktoren mithilfe trigonometrischer Identit&#xE4;ten umzuschreiben.
Ist die Potenz der Sekans gerade, wird der Tangens als Substitutionsvariable gew&#xE4;hlt. Da die Ableitung des Tangens gleich der Sekans zum Quadrat ist, kann ein Faktor sec^2 x aus dem Integranden abgetrennt werden. Die verbleibende gerade Potenz der Sekans wird dann vollst&#xE4;ndig in Tangens ausgedr&#xFC;ckt, indem die Identit&#xE4;t sec^2 x = 1 + tan^2 x verwendet wird. Dadurch wird das Integral zu einem Polynom in der neuen Variablen, das mit algebraischen Standardmethoden integriert werden kann. Nach der Integration wird das Ergebnis wieder in Abh&#xE4;ngigkeit von der urspr&#xFC;nglichen Variablen dargestellt.
Ist die Potenz des Tangens ungerade, wird die Sekans als Substitutionsvariable verwendet. In diesem Fall wird ein Faktor der Form sec x tan x aus dem Integranden abgetrennt, was der Ableitung der Sekans entspricht. Die verbleibende gerade Potenz des Tangens wird mithilfe der Identit&#xE4;t tan^2x = sec^2x &#x2212; 1 in Terme der Sekans umgeschrieben. Diese Substitution transformiert das Integral erneut in einen Polynomausdruck, der direkt integriert werden kann.
In beiden F&#xE4;llen vereinfacht die Verwendung geeigneter Substitutionen und Identit&#xE4;ten das urspr&#xFC;ngliche trigonometrische Integral zu einer gut handhabbaren Polynomform, sodass eine direkte Auswertung und anschlie&#xDF;ende R&#xFC;cksubstitution in trigonometrische Funktionen m&#xF6;glich ist.

Integrale von Potenzen des Tangens und der Sekans

Integrale, die Potenzen des Tangens und der Sekans enthalten, werden &#xFC;blicherweise durch Substitution berechnet, wobei die Vorgehensweise von der ...

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

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Integrals of Powers of Secant and Tangent

solving secant and tangent integrals

Integrals of secant and tangent powers are often solved by choosing a trig substitution that matches the integrand. The key idea is to use the parity of the exponents, which means whether a power is even or odd. A good substitution lets part of the integrand pair with the derivative of a trig function.

When the power of secant is even, tangent is the better substitution variable. Since the derivative of tangent is secant squared, a factor of sec^2 x can be split off. The remaining even power of secant is then rewritten with the identity sec^2 x = 1 + tan^2 x.

This turns the integral into a polynomial in tangent. Polynomial forms are easier to integrate with standard algebra steps. After the integral is found, the answer is changed back into the original trig variable.

When the power of tangent is odd, secant is used instead. In that case, a factor of sec(x)tan(x) is separated from the integrand. This matches the derivative of secant, and the remaining even power of tangent is rewritten with tan^2 x = sec^2 x - 1.

That substitution also changes the problem into a polynomial expression. The polynomial can then be integrated directly. Finally, the result is written again in terms of the original trigonometric functions.

Solving Secant and Tangent Integrals

integration by parts: indefinite integrals

integration-by-parts-indefinite-integrals

Integration by Parts in AC Circuit Analysis

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integration by parts: definite integrals

Using Parts and Substitution for Area

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Integration by parts can help evaluate piecewise integrals that appear in signal processing. In smart speakers, audio inputs are modeled as piecewise functions and analyzed with trigonometric functions such as cosine. This lets complex sound waves be broken into simpler frequency parts. When a definite integral uses a piecewise function, the integral is split into separate parts. One part may contain a constant times cosine, which can be found with basic integration rules. Another part may...

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Solving Trig Integrals with Odd and Even Powers

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integrals of powers of secant and tangent

Integrals of secant and tangent powers are often solved by choosing a trig substitution that matches the integrand. The key idea is to use the parity of the exponents, which means whether a power is even or odd. A good substitution lets part of the integrand pair with the derivative of a trig function. When the power of secant is even, tangent is the better substitution variable. Since the derivative of tangent is secant squared, a factor of sec^2 x can be split off. The remaining even power...

trigonometric substitution

Finding Ellipse Area with Trig Substitution

Trigonometric substitution is a method for solving integrals that contain square root expressions from quadratic forms. It works well when the integrand looks like a familiar geometric equation, such as a circle or an ellipse. Here, the method is applied to estimate the area inside a Molniya satellite orbit. Molniya satellites move in highly elliptical orbits and repeatedly sweep across the same regions of space as they go around Earth. The orbit is modeled as an ellipse in standard form, and...

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Riemann Sum Area Estimates

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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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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improper integrals: discontinuous integrands

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Improper Integrals at an Endpoint

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key terms and concepts

Learn how to solve integrals with secant and tangent powers by using trig substitutions, identities, and parity rules that turn them into polynomials.

Solving Secant and Tangent Integrals | 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

Solving Secant and Tangent Integrals

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