Name: solving trig integrals with odd and even powers
Uploaded: 2026-01-12T12:24:10.000Z
Duration: 1 min 29 s
Channel: JoVE

Lors de la résolution des intégrales impliquant le produit de sinus élevé à la puissance m et du cosinus élevé à la puissance n, l’approche dépend du fait que les exposants soient impaires ou pairs.Si l’une des puissances m ou n est impaire, un processus en trois étapes est utilisé. Premièrement, un terme est extrait de la fonction impaire. Ensuite, le terme à puissance paire restant est réécrit en utilisant une identité trigonométrique en fonction de l’autre fonction. Enfin, la fonction cosinus est choisie comme u, et sa dérivée est trouvée. La substitution transforme l’intégrale en une expression polynomiale, qui est ensuite intégrée puis finalement convertie en termes trigonométriques.Si les deux pouvoirs sont égaux, des identités demi-angles sont appliquées. Le sinus et le cosinus au carré sont remplacés par les identités, résolvant l’intégrale étape par étape en une forme plus simple, ce qui facilite la résolution.Ces concepts simplifient les calculs mathématiques dans les applications d’ingénierie. Par exemple, dans un circuit AC purement résistif, la puissance moyenne consommée par une résistance est déterminée en intégrant tension et courant. L’intégrale implique des puissances paires, donc l’identité du demi-angle est utilisée pour la résoudre.

intégrales de puissances de sinus et de cosinus

Les int&#xE9;grales trigonom&#xE9;triques consistent &#xE0; int&#xE9;grer des expressions contenant des puissances de sinus, de cosinus et de fonctions apparent&#xE9;es. Elles sont fr&#xE9;quentes dans les probl&#xE8;mes d&#x2019;analyse et trouvent des applications en physique et en ing&#xE9;nierie. La m&#xE9;thode pour int&#xE9;grer des expressions de la forme sin^m(x)cos^n(x) d&#xE9;pend de la parit&#xE9; des exposants.
Si la puissance du sinus est impaire, vous extrayez un facteur sinus de l&#x2019;int&#xE9;grande, en laissant une puissance paire de sinus. Les termes de sinus restants sont ensuite r&#xE9;&#xE9;crits en fonction du cosinus gr&#xE2;ce &#xE0; l&#x2019;identit&#xE9; pythagoricienne. Vous choisissez alors le cosinus comme variable de substitution, et le facteur sinus isol&#xE9; se combine avec dx pour former la diff&#xE9;rentielle correspondante. L&#x2019;int&#xE9;grale se transforme ainsi en une int&#xE9;grale polynomiale, qui peut &#xEA;tre int&#xE9;gr&#xE9;e directement.
Si la puissance du cosinus est impaire, vous utilisez une approche similaire. Vous factorisez un facteur cosinus, en laissant une puissance paire de cosinus. Les termes en cosinus restants sont r&#xE9;&#xE9;crits en fonction du sinus, et vous choisissez le sinus comme variable de substitution. Le cosinus factoris&#xE9; fournit la diff&#xE9;rentielle, transformant l&#x2019;int&#xE9;grale en une int&#xE9;grale polynomiale que vous pouvez r&#xE9;soudre.
Lorsque les deux puissances sont paires, la substitution directe n&#x2019;est pas efficace. Dans ce cas, vous utilisez les identit&#xE9;s de demi-angle pour r&#xE9;duire les puissances et simplifier l&#x2019;int&#xE9;grande avant l&#x2019;int&#xE9;gration. En r&#xE9;sum&#xE9;, les int&#xE9;grales trigonom&#xE9;triques impliquant des produits de sinus et de cosinus se traitent par substitution lorsqu&#x2019;un exposant est impair, et par des identit&#xE9;s de demi-angle lorsque les deux exposants sont pairs.
En bref, les int&#xE9;grales trigonom&#xE9;triques impliquant des puissances de sinus et de cosinus se traitent en choisissant une substitution selon qu&#x2019;un exposant est impair ou que les deux exposants sont pairs. Si une puissance est impaire, vous utilisez la substitution ; si les deux sont paires, les identit&#xE9;s de demi-angle simplifient l&#x2019;int&#xE9;grale.

Intégrales de puissances de sinus et de cosinus

Les int&#xE9;grales trigonom&#xE9;triques consistent &#xE0; int&#xE9;grer des expressions contenant des puissances de sinus, de cosinus et de fonctions ...

Education

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Calculus

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Techniques of Integration

Techniques d'intégration

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

Integrals of Powers of Sine and Cosine

solving trig integrals with odd and even powers

Trigonometric integrals are solved by looking at the powers of sine and cosine in the expression. These integrals appear often in calculus, and they can also be useful in physics and engineering. For integrals of the form sin^m(x)cos^n(x), the first step is to check whether the sine power or the cosine power is odd.

If the sine power is odd, one sine factor is separated from the integrand. That leaves an even power of sine. The remaining sine terms are rewritten using the Pythagorean identity, and cosine is used as the substitution variable. The separated sine factor then combines with dx to form the differential, which turns the integral into a polynomial that can be integrated directly.

If the cosine power is odd, the same idea is used with cosine instead. One cosine factor is factored out, leaving an even power of cosine. The remaining cosine terms are rewritten in terms of sine, and sine is chosen as the substitution variable. The factored cosine provides the differential, so the integral becomes a solvable polynomial.

When both powers are even, direct substitution does not work well. In that case, half-angle identities are used to reduce the powers and simplify the integrand before integration. In summary, trigonometric integrals with powers of sine and cosine are handled by substitution when one exponent is odd. When both exponents are even, half-angle identities provide the needed simplification.

Solving Trig Integrals with Odd and Even Powers

solving-trig-integrals-with-odd-and-even-powers

integration by parts: indefinite integrals

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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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integrals of powers of sine and cosine

Trigonometric integrals are solved by looking at the powers of sine and cosine in the expression. These integrals appear often in calculus, and they can also be useful in physics and engineering. For integrals of the form sin^m(x)cos^n(x), the first step is to check whether the sine power or the cosine power is odd. If the sine power is odd, one sine factor is separated from the integrand. That leaves an even power of sine. The remaining sine terms are rewritten using the Pythagorean identity,...

integrals of powers of secant and tangent

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...

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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...

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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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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: infinite intervals

Improper Integrals and Exponential Decay

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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 trigonometric integrals with powers of sine and cosine by using substitution when one exponent is odd and half-angle identities when both are even.

Solving Trig Integrals with Odd and Even Powers | 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 Trig Integrals with Odd and Even Powers

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