Example Question - integration 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.
Integration of a Polynomial
Certainly! You are asked to integrate the given polynomial. Here’s how you do it: The integral of a polynomial is found by integrating each term individually. The integral of \( x^n \) with respect to \( x \) is \( \frac{x^{n+1}}{n+1} \) provided \( n \neq -1 \). Let’s integrate each term of the given polynomial: \[ \int (8x^3 - x^2 + 5x - 1) dx \] Integrating term by term: \[ = 8 \int x^3 dx - \int x^2 dx + 5 \int x dx - \int 1 dx \] Applying the power rule for integration: \[ = 8 \cdot \frac{x^{3+1}}{3+1} - \frac{x^{2+1}}{2+1} + 5 \cdot \frac{x^{1+1}}{1+1} - x + C \] Simplifying the expressions: \[ = 8 \cdot \frac{x^4}{4} - \frac{x^3}{3} + 5 \cdot \frac{x^2}{2} - x + C \] Further simplifying, which also involves reducing the fraction \(8 \cdot \frac{x^4}{4}\) to \(2x^4\): \[ = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] So the integral of \( 8x^3 - x^2 + 5x - 1 \) with respect to \( x \) is \( 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \), where \( C \) is the constant of integration.
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