Example Question - polynomial function integration
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Integrating a Polynomial Function
Claro, podemos resolver la integral de la función f(x) = 3x^2 - 2 con respecto a x. La integral indefinida se obtiene al encontrar la antiderivada de f(x). En este caso, necesitamos aplicar la regla de potencias para la integración, que dice que ∫x^n dx = (x^(n+1))/(n+1) + C, donde C es la constante de integración. Vamos a integrar la función término por término: ∫(3x^2 - 2) dx = 3∫x^2 dx - ∫2 dx Ahora aplicaremos la regla de potencias mencionada anteriormente: Para 3∫x^2 dx, n es igual a 2, así que la antiderivada será (x^(2+1))/(2+1), que se simplifica a (x^3)/3. Para ∫2 dx, simplemente tratamos 2 como una constante multiplicando a x^0, por lo que su antiderivada es 2x (dado que la antiderivada de x^0 es x). Por lo tanto: 3∫x^2 dx = 3 * (x^3)/3 = x^3 (la constante 3 se cancela con el denominador 3). ∫2 dx = 2x Sumamos las antiderivadas para obtener la integral indefinida completa: ∫(3x^2 - 2) dx = x^3 - 2x + C Donde C es la constante de integración que no se conoce a menos que se den más condiciones o límites para la integral.
Integration of a Polynomial Function
Sure, the given integral is: ∫ (8x^3 - x^2 + 5x - 1) dx We need to integrate each term separately. The integral of a polynomial function is found by increasing the exponent by one and dividing by the new exponent. Here's how it works for each term: 1. Integral of 8x^3 dx: Increase exponent by 1 (from 3 to 4), then divide by the new exponent (4). ∫ 8x^3 dx = 8/4 x^4 = 2x^4 2. Integral of -x^2 dx: Increase exponent by 1 (from 2 to 3), then divide by the new exponent (3). ∫ -x^2 dx = -1/3 x^3 3. Integral of 5x dx: Increase exponent by 1 (from 1 to 2), then divide by the new exponent (2). ∫ 5x dx = 5/2 x^2 4. Integral of -1 dx: Since the exponent is 0 (because -1 is the same as -1x^0), we just multiply x by the constant. ∫ -1 dx = -1 * x = -x Now, let's put it all together: ∫ (8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Don't forget to add the constant of integration, C, at the end. The final answer is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
Integration of Polynomial Function
Certainly! You need to integrate the given polynomial function with respect to x. The integration of a polynomial function is done term by term. Here are the steps: Given function: ∫(8x^3 - x^2 + 5x - 1)dx 1. Integrate each term separately using the power rule for integration: ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. For the first term 8x^3: ∫8x^3 dx = 8 * (x^(3+1))/(3+1) = 8 * x^4/4 = 2x^4 For the second term -x^2: ∫(-x^2) dx = - (x^(2+1))/(2+1) = -x^3/3 For the third term 5x: ∫5x dx = 5 * (x^(1+1))/(1+1) = 5/2 x^2 For the fourth term -1: ∫(-1) dx = -x 2. Combine the integrated terms and include the constant of integration: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Therefore, the indefinite integral of the given function 8x^3 - x^2 + 5x - 1 with respect to x is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
Integration of a Polynomial Function
To solve the given integral, we will integrate each term separately: \[ \int (8x^3 - x^2 + 5x - 1)\ dx \] We integrate each term using the power rule for integration, which is: \[ \int x^n\ dx = \frac{x^{n+1}}{n+1} + C \] where \(n \neq -1\) and \(C\) is the constant of integration. Now, integrating each term: 1. \(\int 8x^3\ dx = \frac{8}{4}x^{3+1} = 2x^4\) 2. \(\int -x^2\ dx = -\frac{1}{3}x^{2+1} = -\frac{1}{3}x^3\) 3. \(\int 5x\ dx = \frac{5}{2}x^{1+1} = \frac{5}{2}x^2\) 4. \(\int -1\ dx = -x\) Combine these results and add the constant of integration (\(C\)): \[ \int (8x^3 - x^2 + 5x - 1)\ dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] So, the integral of \(8x^3 - x^2 + 5x - 1\) with respect to \(x\) is: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
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