Example Question - antiderivative of polynomial
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Integration of Polynomial
The given integral is: ∫ (8x^3 - x^2 + 5x - 1) dx To solve this integral, we integrate each term of the polynomial separately: ∫8x^3 dx = (8/4)x^4 = 2x^4 ∫-x^2 dx = -(1/3)x^3 = -x^3/3 ∫5x dx = (5/2)x^2 = 5x^2/2 ∫-1 dx = -x Combining the integrated terms gives us the antiderivative: 2x^4 - x^3/3 + 5x^2/2 - x + C where C is the constant of integration.
Integral of a Polynomial Expression
Certainly! The given expression is an integral that you need to evaluate. You have the following expression: ∫(8x^3 - x^2 + 5x - 1) dx To solve this, you integrate each term separately with respect to x. The integral of a sum or difference of functions is the sum or difference of their integrals. Here's how to integrate each term: 1. ∫8x^3 dx = (8/4)x^(3+1) = 2x^4 2. ∫-x^2 dx = -(1/3)x^(2+1) = -(1/3)x^3 3. ∫5x dx = (5/2)x^(1+1) = (5/2)x^2 4. ∫-1 dx = -x Now combine these results to get the complete antiderivative: ∫(8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Here, C represents the constant of integration, which is a standard addition in indefinite integrals, as there are an infinite number of antiderivatives differing by a constant.
Solving Integral of a Polynomial Function
The image shows an integral that you'd like to solve: \[ \int (8x^3 - x^2 + 5x - 1) \, dx \] To solve this integral, integrate each term individually with respect to \( x \). When integrating a polynomial term of the form \( x^n \), the antiderivative is \( \frac{x^{n+1}}{n+1} \) provided that \( n \neq -1 \). \[ \int 8x^3 \, dx = \frac{8}{4}x^{4} = 2x^4 \] \[ \int (-x^2) \, dx = -\frac{1}{3}x^{3} \] \[ \int 5x \, dx = \frac{5}{2}x^{2} \] \[ \int (-1) \, dx = -x \] Now summing all of the individual antiderivatives and adding a constant of integration \( C \), we have: \[ 2x^4 - \frac{1}{3}x^{3} + \frac{5}{2}x^{2} - x + C \] So the integral of \( 8x^3 - x^2 + 5x - 1 \) with respect to \( x \) is: \[ \int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^{3} + \frac{5}{2}x^{2} - x + C \]
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