Name: using parts and substitution for area
Uploaded: 2026-01-12T12:24:42.000Z
Duration: 1 min 23 s
Channel: JoVE

Las integrales definidas de productos de dos funciones sobre intervalos fijos pueden resolverse mediante integración por partes.Para resolver el lado derecho de esta expresión, se evaluará la diferencia del producto de funciones entre los extremos del intervalo, y el término restante se tratará como una integral definida.Un ejemplo útil es la integral de la función tangente inversa. Como no existe una fórmula integral estándar para esta función, el integrando se trata en cambio como un producto de la tangente inversa y la constante 1.La tangente inversa se toma como la función a diferenciar y la constante se integra.Sustituyendo en la fórmula de integración por partes, el primer término puede resolverse evaluando directamente el producto en los puntos finales. La integral restante puede resolverse entonces por sustitución.Una nueva variable, t, se establece igual a 1 más x al cuadrado y se ajustan los límites de integración. La integral entonces se convierte en una expresión recíproca que se simplifica a una forma logarítmica.El logaritmo natural de uno es igual a cero, por lo que la expresión final se simplifica para mostrar el área bajo la curva entre los dos límites.

integración por partes: integrales definidas

Integration by Parts: Definite Integrals

Las integrales definidas que implican el producto de dos funciones en un intervalo fijo se pueden evaluar mediante la integraci&#xF3;n por partes. Este m&#xE9;todo reescribe la integral como la diferencia de un producto calculado en los extremos y una integral definida restante, que suele ser m&#xE1;s sencilla de calcular.
Un ejemplo representativo es la integral definida de la funci&#xF3;n arcotangente. Dado que no existe una f&#xF3;rmula de integraci&#xF3;n directa para la arcotangente (x), el integrando se reescribe como un producto de la arcotangente (x) y la funci&#xF3;n constante 1. Se elige la arcotangente para la diferenciaci&#xF3;n, mientras que la constante se integra, lo que simplifica la estructura del problema.
Sustituir estas elecciones en la f&#xF3;rmula de integraci&#xF3;n por partes permite evaluar directamente el primer t&#xE9;rmino calculando el producto en los extremos del intervalo. El t&#xE9;rmino restante es una integral definida que se puede resolver mediante sustituci&#xF3;n. Al introducir una nueva variable igual a 1 + x^2, la integral se transforma en una forma rec&#xED;proca que se integra como una expresi&#xF3;n logar&#xED;tmica, con los l&#xED;mites ajustados seg&#xFA;n corresponda.
Evaluar el t&#xE9;rmino logar&#xED;tmico en los l&#xED;mites simplifica el resultado, ya que el logaritmo natural de 1 es cero. La expresi&#xF3;n final representa el &#xE1;rea bajo la curva de la arcotangente entre los l&#xED;mites dados, lo que ilustra c&#xF3;mo la integraci&#xF3;n por partes puede aplicarse de manera eficaz a integrales definidas que involucran funciones sin primitiva en t&#xE9;rminos elementales.

Integración por partes: Integrales definidas

Las integrales definidas que implican el producto de dos funciones en un intervalo fijo se pueden evaluar mediante la integraci&#xF3;n por partes. Este ...

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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, while the constant is chosen to integrate, which simplifies the setup.

After the integration by parts formula is applied, the first term is found by evaluating the product at the interval endpoints. The remaining definite integral is then handled with substitution. Letting the new variable equal 1 + x2 changes the integral into a reciprocal form, which integrates to a logarithmic expression.

The limits must be adjusted to match the new variable. When the logarithmic term is evaluated at the bounds, the natural logarithm of one becomes zero. The final result gives the area under the inverse tangent curve between the given limits and shows how integration by parts can work well for definite integrals without elementary antiderivatives.

Using Parts and Substitution for Area

integration by parts: indefinite integrals

integration-by-parts-indefinite-integrals

Integration by Parts in AC Circuit Analysis

Integration by parts is a calculus method for handling integrals that contain a product of two functions. It is useful when direct integration is not practical. The method comes from the product rule for differentiation, which says that the derivative of a product is the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating that product-rule identity and rearranging the terms, the integration by parts formula is obtained.

integration by parts: definite integrals

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,...

integration by parts: problem solving

Integration by Parts for Signal Processing

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...

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,...

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...

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...

integration of rational functions using partial fractions

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

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...

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...

approximate integration

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...

trapezoidal rule

Estimating Distance from Velocity Data

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...

simpson's rule ii

Simpson's Rule for Curved Sheet Area

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improper integrals: discontinuous integrands

improper-integrals-discontinuous-integrands

Improper Integrals at an Endpoint

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key terms and concepts

Learn how integration by parts and substitution simplify a definite integral of arctan x. The method turns the problem into endpoint values and a logarithm.

Using Parts and Substitution for Area | 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

Using Parts and Substitution for Area

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