Example Question - polynomial integration rule
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Integration of Polynomial Functions
To solve the given integral, we will integrate each term separately. The integral given is: ∫(8x^3 - x^2 + 5x - 1)dx To integrate this, we use the power rule for integration: ∫x^n dx = x^(n+1) / (n+1) + C, where n ≠ -1 and C is the constant of integration. Applying this rule for each term, we get: ∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1) / (3+1)) + C_1 = 8 * (x^4 / 4) + C_1 = 2x^4 + C_1 ∫(-x^2) dx = - ∫x^2 dx = - (x^(2+1) / (2+1)) + C_2 = - (x^3 / 3) + C_2 ∫5x dx = 5 * ∫x dx = 5 * (x^(1+1) / (1+1)) + C_3 = 5 * (x^2 / 2) + C_3 = (5/2)x^2 + C_3 ∫(-1) dx = -x + C_4 Now, adding them all together and combining the constants of integration, we get: ∫(8x^3 - x^2 + 5x - 1)dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Where C is the constant of integration, representing that there can be any constant added to the integral (C = C_1 + C_2 + C_3 + C_4).
Integration of Polynomials
To solve the given integral, integrate each term with respect to x: \[ \int (8x^3 - x^2 + 5x - 1) \, dx \] Integrate term by term: \[ = \int 8x^3 \, dx - \int x^2 \, dx + \int 5x \, dx - \int 1 \, dx \] Now apply the power rule of integration which is: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{where} \quad n \neq -1 \] So we get: \[ = \frac{8x^{3+1}}{3+1} - \frac{x^{2+1}}{2+1} + \frac{5x^{1+1}}{1+1} - x + C \] Simplify the expression: \[ = \frac{8x^4}{4} - \frac{x^3}{3} + \frac{5x^2}{2} - x + C \] \[ = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] Here, \( C \) is the constant of integration. So the final answer is: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
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