Example Question - integrating polynomial equations
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Integral of a Polynomial Function
The integral in the image is: ∫ (8x^3 - x^2 + 5x - 1) dx To solve the integral, integrate each term separately with respect to x: ∫ 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 Now, combine the integrated terms: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Here, C is the constant of integration. The final answer for the integral is: 2x^4 - (x^3/3) + (5x^2/2) - 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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