Example Question - integrating polynomials
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Integrating Polynomials Using Power Rule
To solve the integral provided in the image: ∫(8x^3 - x^2 + 5x - 1)dx You need to integrate each term separately using the power rule for integration. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n ≠ -1. Applying the power rule to each term: For 8x^3, ∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1))/(3+1) = 8 * (x^4)/4 = 2x^4 For -x^2, ∫(-x^2) dx = - ∫x^2 dx = - (x^(2+1))/(2+1) = - (x^3)/3 For 5x, ∫5x dx = 5 * ∫x dx = 5 * (x^(1+1))/(1+1) = 5 * (x^2)/2 = (5/2)x^2 For -1, ∫(-1) dx = - ∫1 dx = -x Now, combine all the integrated terms: ∫(8x^3 - x^2 + 5x - 1)dx = 2x^4 - (x^3)/3 + (5/2)x^2 - x + C Where C is the constant of integration.
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: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
Polynomial Integral by Term
The integral presented in the image is of a polynomial. Integrating polynomials term by term, we find: ∫(8x^3 - x^2 + 5x - 1) dx We will integrate each term separately using the power rule for integrals. For a general term ax^n, the integral is (a/(n+1))x^(n+1), plus a constant of integration which we'll add at the end. Applying this rule to each term: ∫8x^3 dx = (8/4)x^(3+1) = 2x^4 ∫-x^2 dx = (-1/3)x^(2+1) = -1/3 x^3 ∫5x dx = (5/2)x^(1+1) = 5/2 x^2 ∫-1 dx = -x Adding these results together and including the constant of integration C, we get the antiderivative: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
Solving Indefinite Integral of a Polynomial Function
The question in the image is asking for the integral of the given function with respect to x: ∫(8x^3 - x^2 + 5x - 1) dx To solve this indefinite integral, you would integrate each term separately. The integral of a sum is the sum of the integrals, so you can integrate each term individually. The integral of 8x^3 with respect to x is (8/4)x^(3+1) = 2x^4. The integral of -x^2 with respect to x is -(1/3)x^(2+1) = -1/3 x^3. The integral of 5x with respect to x is (5/2)x^(1+1) = 5/2 x^2. The integral of -1 with respect to x is -x. Putting it all together, the indefinite integral of the function is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C where C is the constant of integration.
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