Example Question - definite integral
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Definite Integral of the Square of Secant Function
<p>La función secante al cuadrado, \( \sec^2(x) \), es la derivada de la función tangente, \( \tan(x) \). Por lo tanto, para resolver esta integral definida, podemos aplicar la regla fundamental del cálculo que dice que la integral de la derivada de una función en un intervalo es igual a la diferencia de los valores de la función en los extremos del intervalo:</p> <p>\[ \int_{0}^{4} \sec^2 x \, dx = \tan(x) \Big|_{0}^{4} \]</p> <p>Calculamos los valores de \( \tan(x) \) en \( x = 4 \) y \( x = 0 \) y encontramos la diferencia:</p> <p>\[ = \tan(4) - \tan(0) \]</p> <p>Sin embargo, notarás que hay un problema al querer evaluar \( \tan(4) \), ya que 4 radianes está muy cerca de \( \frac{\pi}{2} \approx 1.5708 \) y la función tangente no está definida en \( \frac{\pi}{2} + n\pi \), donde \( n \) es cualquier número entero. Al sustituir \( x = 4 \) en la función tangente, obtendremos un valor no definido, lo que significa que la integral no converge en ese punto.</p> <p>Por ende, debido a la discontinuidad en \( x = 4 \), no podemos obtener un valor numérico real para esta integral definida con los límites dados de \( 0 \) a \( 4 \). Si se cometió un error al escribir los límites de integración y se trataba de un intervalo donde la función tangente es continua, los pasos a seguir para resolverla serían correctos.</p>
Definite Integral of a Quadratic Function
Para resolver la integral definida, primero encontraremos la antiderivada de \( x^2 + x - 9 \). <p>\( \int x^2 + x - 9 \, dx = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 9x + C \)</p> Ahora evaluaremos esta antiderivada en los límites superior e inferior de la integral. <p>\( \left[ \frac{1}{3}x^3 + \frac{1}{2}x^2 - 9x \right]_{5}^{10} \)</p> <p>\( = \left( \frac{1}{3}(10)^3 + \frac{1}{2}(10)^2 - 9(10) \right) - \left( \frac{1}{3}(5)^3 + \frac{1}{2}(5)^2 - 9(5) \right) \)</p> <p>\( = \left( \frac{1}{3}(1000) + \frac{1}{2}(100) - 90 \right) - \left( \frac{1}{3}(125) + \frac{1}{2}(25) - 45 \right) \)</p> <p>\( = \left( 333\frac{1}{3} + 50 - 90 \right) - \left( 41\frac{2}{3} + 12\frac{1}{2} - 45 \right) \)</p> <p>\( = 293\frac{1}{3} - 8\frac{5}{6} \)</p> <p>\( = 293\frac{1}{3} - 8\frac{10}{12} \)</p> <p>\( = 293\frac{4}{12} - 8\frac{10}{12} \)</p> <p>\( = 284\frac{6}{12} \)</p> <p>\( = 284\frac{1}{2} \)</p> La integral definida de \( x^2 + x - 9 \) de 5 a 10 es \( 284\frac{1}{2} \) o \( 284.5 \).
Solving a Definite Integral Involving Exponential and Power Functions
<p>To solve the integral \(\int_0^{1/2} x^{1/2}e^{2x} \, dx\), we can use the method of integration by parts, which states that \(\int u \, dv = uv - \int v \, du\).</p> <p>Let \( u = x^{1/2} \) and \( dv = e^{2x} \, dx \).</p> <p>Then we have \( du = \frac{1}{2}x^{-1/2} \, dx \) and \( v = \frac{1}{2}e^{2x} \).</p> <p>The integral becomes:</p> <p>\( \int_0^{1/2} x^{1/2}e^{2x} \, dx = \left. \frac{1}{2}x^{1/2}e^{2x} \right|_0^{1/2} - \int_0^{1/2} \frac{1}{2}e^{2x} \frac{1}{2}x^{-1/2} \, dx \)</p> <p>\( = \left. \frac{1}{2}x^{1/2}e^{2x} \right|_0^{1/2} - \frac{1}{4} \int_0^{1/2} x^{-1/2}e^{2x} \, dx \)</p> <p>To solve the remaining integral, we use integration by parts again with \( u = x^{-1/2} \) and \( dv = e^{2x} \, dx \).</p> <p>Then we get \( du = -\frac{1}{2}x^{-3/2}dx \) and \( v = \frac{1}{2}e^{2x} \).</p> <p>The remaining integral is:</p> <p>\( \frac{1}{4} \left( \left. x^{-1/2}e^{2x} \right|_0^{1/2} - \int_0^{1/2} \frac{1}{2}e^{2x}(-\frac{1}{2})x^{-3/2} \, dx \right) \)</p> <p>\(= \frac{1}{4} \left( \left. x^{-1/2}e^{2x} \right|_0^{1/2} + \frac{1}{4} \int_0^{1/2} x^{-3/2}e^{2x} \, dx \right) \)</p> <p>The evaluation of these integrals at the limits \(0\) and \(\frac{1}{2}\) must be approached with caution, because the terms involving \(x^{-1/2}\) and \(x^{-3/2}\) are undefined at \(x=0\). These expressions suggest the integral does not converge at the lower limit, however, this is dependent on the behavior of the exponential function as it approaches zero, which could negate the potential divergence caused by the power of \(x\).</p> <p>Without performing a limit analysis, the solution remains indeterminate at \(x = 0\). A detailed analysis might involve considering the limit of the integrand as \(x\) approaches zero and applying L'Hôpital's rule if necessary.</p>
Finding the Integral of a Quadratic Function
<p>\(\int f(x) \,dx = \int x^2 \,dx\)</p> <p>\(= \frac{x^{2+1}}{2+1} + C\)</p> <p>\(= \frac{x^3}{3} + C\)</p>
Solving a Definite Integral
La integral que se muestra en la imagen es: ∫₀ᵃ xdx Para resolver esta integral, aplicaremos la regla de la potencia para la integración. La regla de la potencia nos dice que la integral de x^n dx es (x^(n+1))/(n+1) + C, donde C es la constante de integración. En este caso, n = 1. Por lo tanto: ∫ xdx = (x^(1+1))/(1+1) + C ∫ xdx = x^2/2 + C Ahora, dado que estamos buscando la integral definida desde 0 hasta a, evaluamos la expresión x^2/2 en a y 0, y restamos el valor menor del valor mayor: ∫₀ᵃ xdx = (a^2/2) - (0^2/2) ∫₀ᵃ xdx = a^2/2 - 0 ∫₀ᵃ xdx = a^2/2 Como no existe un término (a^2/2) entre las opciones dadas, debemos encontrar la opción que es matemáticamente equivalente. Dado que la integral se está tomando en un intervalo donde a es el límite superior, el resultado no puede ser negativo. Al examinar las opciones, vemos que la opción c) a²/2 es la equivalente a nuestro resultado. Por lo tanto, la respuesta correcta es: c) a²/2
Solving Definite Integral of x
Para resolver la integral definida \(\int_{0}^{a} x dx\), primero encontraremos la integral indefinida de \(x\) con respecto a \(x\), y después evaluaremos esa integral entre los límites de 0 a \(a\). La integral indefinida de \(x\) es \(\frac{x^2}{2}+C\), donde \(C\) es la constante de integración. Dado que estamos haciendo una integral definida, la constante \(C\) se cancelará al evaluar en los límites, así que podemos ignorarla. Ahora evaluamos la integral indefinida entre los límites dados: \[ \left[\frac{x^2}{2}\right]_0^a = \frac{a^2}{2} - \frac{0^2}{2} = \frac{a^2}{2} - 0 = \frac{a^2}{2} \] Por lo tanto, la solución de la integral es \(\frac{a^2}{2}\). La respuesta correcta es la opción c) \(\frac{a^2}{2}\).
Solving a Definite Integral from 0 to a of x with Respect to x
Para resolver la integral indicada en la imagen, vamos a calcular la integral de x respecto a x desde 0 hasta a. La integral de x con respecto a x es: ∫ x dx => (1/2) x^2 + C Donde C es la constante de integración. Sin embargo, como estamos evaluando la integral definida de 0 a a, no necesitamos preocuparnos por la constante de integración. Evaluemos la integral definida: ∫ de 0 a a x dx = [(1/2) x^2] de 0 a a = (1/2) a^2 - (1/2) 0^2 = (1/2) a^2 - 0 = (1/2) a^2 Por lo tanto, la solución a la integral es (1/2) a^2. La opción correcta, de acuerdo a las opciones proporcionadas en la imagen, es la opción d) \( \frac{a^2}{2} \).
Incorrect Integration Solution Provided
Para resolver la integral dada, utilizamos la regla de la potencia para integrales. La integral de x con respecto a x es \(\frac{x^2}{2}\), entonces: \[\int_0^a xdx = \left[\frac{x^2}{2}\right]_0^a\] Esto significa que evaluamos la función \(\frac{x^2}{2}\) en a y restamos el valor de la misma función evaluada en 0. Primero, evaluamos en a: \[ \left[\frac{a^2}{2}\right] - \left[\frac{0^2}{2}\right] = \frac{a^2}{2} - 0 = \frac{a^2}{2} \] Entonces, la solución correcta de la integral es: \[ \frac{a^2}{2} \] Mirando las opciones dadas en la pregunta, la respuesta es la opción c) \( -\frac{a^2}{2} \). Parece haber un error aquí; la evaluación de la integral es positiva dado que es el área bajo la curva de x desde 0 hasta a, por lo tanto, la respuesta correcta debería ser \(\frac{a^2}{2}\), pero esta opción no está listada correctamente. La opción que más se aproxima es la opción d) \(\frac{a^2}{3}\), pero esta tampoco es la respuesta correcta. Debería haber una opción que sea \(\frac{a^2}{2}\) para ser la solución correcta de la integral proporcionada.
Solving Definite Integral Using Fundamental Theorem of Calculus
Para resolver esta integral definida, aplicaremos el teorema fundamental del cálculo. La integral de x con respecto a x es \( \frac{x^2}{2} \), así que aplicamos los límites de integración: \( \int_{0}^{a} xdx = \left[\frac{x^2}{2}\right]_{0}^{a} \) Evaluamos esta expresión en los límites superior e inferior: \( \left[\frac{x^2}{2}\right]_{0}^{a} = \frac{a^2}{2} - \frac{0^2}{2} \) \( = \frac{a^2}{2} - 0 \) \( = \frac{a^2}{2} \) Por lo tanto, la respuesta correcta es la opción \( \text{d) } \frac{a^2}{2} \).
Evaluating a Definite Integral
The quality of the image isn't perfect, but I can make out that you're looking to evaluate the definite integral: \(\int_{0}^{2} (2x) dx\) To solve this definite integral, you can follow the steps below: 1. Find the antiderivative (indefinite integral) of the integrand function f(x) = 2x: \[ F(x) = \int 2x dx = x^2 + C \] where C represents the constant of integration. However, since we're evaluating a definite integral, we can ignore C. 2. Plug in the limits of integration into the antiderivative F(x): \[ F(2) - F(0) = (2^2) - (0^2) = 4 - 0 = 4 \] Thus, the value of the given definite integral is 4.
Indefinite Integration of Tan(x)dx within Specified Bounds
The image shows a mathematical problem asking to solve the definite integral of tan(x)dx within the bounds from 0 to pi. However, the image is blurry and I cannot completely confirm the integral bounds. Please confirm or clarify the bounds of the integral so I can provide the correct solution. If the bounds of the integral are indeed from 0 to pi, we run into an issue. Since tan(x) has a vertical asymptote at x = pi/2, where the function becomes undefined (tan(x) approaches positive or negative infinity as x approaches pi/2 from the left or right, respectively), the integral of tan(x) from 0 to pi does not converge to a finite number. Thus, the integral is not defined in the classical sense, but one could approach it in the context of Cauchy principal values or other extended frameworks. Please verify the bounds so that we could proceed with the appropriate solution if necessary.
Average Value Theorem for Integration
为了回答这个问题，我们可以使用积分的平均值定理。这个定理告诉我们，一个连续函数在某个区间的平均值等于它在这个区间上的定积分除以区间的长度。 问题中给出的区间是 [0, 2]，区间长度为 2 - 0 = 2。同时，我们知道函数 g(x) 在区间 [0, x] 上的定积分的值为 3，表示函数 g(x) 和 x 轴之间这段曲线下包围的面积是 3。 积分的平均值 = (函数在区间上的定积分)/(区间的长度) 此处的定积分是给定的面积，即 3，区间长度是 2。所以平均值 g_avg 为： g_avg = 3 / 2 所以平均值 g_avg = 1.5，对应的选项是 (B) \( \frac{3}{2} \)。
Analyzing Riemann Sums for Integral Approximation
To find the Riemann sum that gives the value of the definite integral \(\int_{-2}^{3} (x^2 + 4x) dx\), we should identify which sum corresponds to calculating the integral using left-hand, right-hand, or midpoint Riemann sums. The integral given is from -2 to 3. Let's start by defining a partition of the interval [-2, 3] into n subintervals; this means each subinterval has width Δx. Given the boundaries a = -2 and b = 3, the width of each subinterval is: \[ Δx = \frac{{b - a}}{n} = \frac{{3 - (-2)}}{n} = \frac{5}{n} \] The general x-value for the k-th subinterval can be expressed in three different forms depending on the Riemann sum method: - Left-hand: \( x_k = a + kΔx \) - Right-hand: \( x_k = a + (k+1)Δx \) - Midpoint: \( x_k = a + \left(k + \frac{1}{2}\right)Δx \) We need to find out which option corresponds to using one of these forms. (A) The term inside the summation \( -2 + \left(\frac{3}{2}\frac{k}{n} + \frac{3}{2}\frac{k}{n} \right) \) doesn't seem to fit any of the standard forms for left, right, or midpoint Riemann sums. (B) \( -2 + \frac{5k}{n} \) fits the left-hand rule where \( x_k = -2 + k Δx \) because cooresponds \( Δx = \frac{5}{n} \). (C) \( -2 + \frac{5}{2} + \left( \frac{5k}{n} \right) \) corresponds to the midpoint Riemann sum, where we start at the midpoint of the first interval and add \( k Δx \). (D) \( -2 + \frac{5}{2} \left( \frac{4}{n} + \frac{2}{n}k \right) \) doesn't match any of the standard forms either. Now using the function \( f(x) = x^2 + 4x \), we can express the Riemann sum based on the left-hand rule (which option B represents). The terms inside the summation should be \( f(x_k)Δx \), which would be \( f(-2 + \frac{5k}{n})Δx \). Substitute \( f(x) \) and \( Δx \): \[ f(x_k) = (-2 + \frac{5k}{n})^2 + 4(-2 + \frac{5k}{n}) \] \[ Δx = \frac{5}{n} \] Hence, the sum is \( Σ [(-2 + \frac{5k}{n})^2 + 4(-2 + \frac{5k}{n})]·\frac{5}{n} \), which simplifies to: \[ Σ \frac{5}{n}[(-2 + \frac{5k}{n})^2 + 4(-2 + \frac{5k}{n})] \] \[ Σ \frac{5}{n}[4 + \frac{25k^2}{n^2} - 20\frac{k}{n} - 8 + 20\frac{k}{n} - \frac{20k}{n}] \] \[ Σ \frac{5}{n}[-4 + \frac{25k^2}{n^2}] \] This expression doesn't match any of the options exactly, but it is closest to option (B), which suggests that there may be an error in the transcription of the options or a simplification error. Thus, option (B) most closely resembles the expression we derived for the Riemann sum based on the left-hand rule: \[ \lim_{n \to \infty} \sum_{k=1}^{n} f(-2 + \frac{5k}{n})·\frac{5}{n} \] However, the terms for \( f(-2 + \frac{5k}{n}) \) must be \( (-2 + \frac{5k}{n})^2 + 4(-2 + \frac{5k}{n}) \) to properly represent the left-hand Riemann sum for this function \( (x^2 + 4x) \) over the interval [-2, 3], and these terms should be inside the summation, not outside as shown in option (B). If there is an error in the transcription or presentation of the options, it might be wise to revisit the original problem to ensure that option (B) was copied correctly. Given the provided choices, none match the correct Riemann sum for the integral exactly, but the form of (B) suggests that it's meant to be the left endpoint approximation.
Solving an Arcsecant Integral
The integral in the image is: \[ \int_0^{\frac{1}{2}} \frac{dx}{\sqrt{1 - x^2}} \] This integral is the integral of the arcsecant derivative. We can use the substitution \( x = \sin(\theta) \) to solve it: \[ dx = \cos(\theta) d\theta \] When \( x = 0 \), \( \sin(\theta) = 0 \), so \( \theta = 0 \). When \( x = \frac{1}{2} \), \( \sin(\theta) = \frac{1}{2} \), which means \( \theta = \frac{\pi}{6} \) since it lies in the first quadrant. Substituting \( x = \sin(\theta) \) into the integral gives us: \[ \int_0^{\frac{\pi}{6}} \frac{\cos(\theta)}{\sqrt{1 - \sin^2(\theta)}} d\theta \] Now, \( \sqrt{1 - \sin^2(\theta)} = \sqrt{\cos^2(\theta)} = \cos(\theta) \) So the integral simplifies to: \[ \int_0^{\frac{\pi}{6}} d\theta \] This is just the integral of \(1\) with respect to \( \theta \) from \(0\) to \(\frac{\pi}{6}\), so it equals \( \theta \) evaluated between these two limits: \[ \theta \Big|_0^{\frac{\pi}{6}} = \frac{\pi}{6} - 0 = \frac{\pi}{6} \] Therefore, the integral equals \( \frac{\pi}{6} \).
Definite Integral of a Function
To solve this definite integral, we need to integrate each term separately within the bounds from -1 to 0. Let's integrate the function \( f(x) = 2x - e^x \). The integral of \( 2x \) is \( x^2 \), and the integral of \( -e^x \) is \( -e^x \), since the derivative of \( e^x \) with respect to \( x \) is \( e^x \). Let's integrate and then apply the bounds: \[ \int_{-1}^{0} (2x - e^x) dx = \left[ x^2 - e^x \right]_{-1}^{0} \] Now, applying the Fundamental Theorem of Calculus, we evaluate this antiderivative at the upper bound x=0 and subtract the value of the antiderivative at the lower bound x=-1: At x=0: \[ x^2 - e^x = 0^2 - e^0 = 0 - 1 = -1 \] At x=-1: \[ x^2 - e^x = (-1)^2 - e^{-1} = 1 - \frac{1}{e} \] Now subtract the value at x=-1 from the value at x=0: \[ (-1)-(1 - \frac{1}{e}) = -1 - 1 + \frac{1}{e} = -2 + \frac{1}{e} \] Hence the value of the definite integral is: \[ \int_{-1}^{0} (2x - e^x) dx = -2 + \frac{1}{e} \]
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