Example Question - antiderivative
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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 an Integral Involving a Trigonometric Function
<p>The integral of \( f(x) = -3 \cos(4x) \) can be found using the standard integration techniques for trigonometric functions.</p> <p>Let \( u = 4x \). Therefore, \( du = 4dx \) or \( \frac{du}{4} = dx \).</p> <p>The integral becomes:</p> <p>\( \int -3 \cos(4x) dx = \int -3 \cos(u) \frac{du}{4} \)</p> <p>\( = -\frac{3}{4} \int \cos(u) du \)</p> <p>\( = -\frac{3}{4} \sin(u) + C \)</p> <p>Substituting back \( u = 4x \):</p> <p>\( = -\frac{3}{4} \sin(4x) + C \)</p> <p>Where \( C \) is the constant of integration.</p>
Finding the Integral of a Function Involving a Fraction with a Variable in the Denominator
<p>\( \int f(x) \, dx = \int \frac{1}{3x} \, dx \)</p> <p>\( = \frac{1}{3} \int \frac{1}{x} \, dx \)</p> <p>\( = \frac{1}{3} \ln|x| + C \)</p> <p>Where \( C \) is the constant of integration.</p>
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.
Solving Integral of Polynomial Function
The image shows an integral that needs to be solved. The expression within the integral is a polynomial function: ∫(8x^3 - x^2 + 5x - 1) dx To solve this, we integrate each term of the polynomial separately with respect to x. The integral of a constant times a power of x is the constant times the fraction of the incremented power of x divided by the new exponent. The integral of a constant is just the constant times x. Here's how this integral is computed: ∫8x^3 dx = 2x^4 (because you add 1 to the exponent and then divide by the new exponent) ∫-x^2 dx = -x^3/3 (same principle) ∫5x dx = 5x^2/2 (again, same principle) ∫-1 dx = -x Putting it all together, the antiderivative of the given function is: 2x^4 - x^3/3 + 5x^2/2 - x + C where C is the constant of integration.
Integration of Polynomials
To solve the integral \[ \int (8x^3 - x^2 + 5x - 1) dx \] we will integrate each term separately. Recall that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\) plus a constant of integration, for any real number \(n\) not equal to -1. Here are the steps: \[ \int 8x^3 dx = \frac{8}{4}x^{3+1} = 2x^4 \] \[ \int (-x^2) dx = -\frac{1}{3}x^{2+1} = -\frac{1}{3}x^3 \] \[ \int 5x dx = \frac{5}{2}x^{1+1} = \frac{5}{2}x^2 \] \[ \int (-1) dx = -x \] Now combine all terms to get the antiderivative: \[ \int (8x^3 - x^2 + 5x - 1) dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] where \(C\) is the constant of integration.
Polynomial Integral by Term
The integral presented in the image is of a polynomial. Integrating polynomials term by term, we find: ∫(8x^3 - x^2 + 5x - 1) dx We will integrate each term separately using the power rule for integrals. For a general term ax^n, the integral is (a/(n+1))x^(n+1), plus a constant of integration which we'll add at the end. Applying this rule to each term: ∫8x^3 dx = (8/4)x^(3+1) = 2x^4 ∫-x^2 dx = (-1/3)x^(2+1) = -1/3 x^3 ∫5x dx = (5/2)x^(1+1) = 5/2 x^2 ∫-1 dx = -x Adding these results together and including the constant of integration C, we get the antiderivative: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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