Example Question - integration of polynomials
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Solving Polynomial Integral with Power Rule
The integral in the image is: ∫(8x^3 - x^2 + 5x - 1)dx To solve this integral, you need to integrate each term separately with respect to x. The integration of a polynomial involves applying the power rule for integration, which says that for any term ax^n, the integral is (a/(n+1))x^(n+1), assuming n ≠ -1. Here's how you integrate each term: For 8x^3: ∫8x^3 dx = (8/4)x^(3+1) = 2x^4 For -x^2: ∫(-x^2) dx = (-1/3)x^(2+1) = -(1/3)x^3 For 5x: ∫5x dx = (5/2)x^(1+1) = (5/2)x^2 For -1: ∫(-1)dx = -x Putting them all together: ∫(8x^3 - x^2 + 5x - 1)dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Here C is the constant of integration, which arises because integration is an indefinite integral.
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.
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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