Example Question - integral 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.
Integral of a Polynomial Expression
Certainly! The given expression is an integral that you need to evaluate. You have the following expression: ∫(8x^3 - x^2 + 5x - 1) dx To solve this, you integrate each term separately with respect to x. The integral of a sum or difference of functions is the sum or difference of their integrals. Here's how to integrate each term: 1. ∫8x^3 dx = (8/4)x^(3+1) = 2x^4 2. ∫-x^2 dx = -(1/3)x^(2+1) = -(1/3)x^3 3. ∫5x dx = (5/2)x^(1+1) = (5/2)x^2 4. ∫-1 dx = -x Now combine these results to get the complete antiderivative: ∫(8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Here, C represents the constant of integration, which is a standard addition in indefinite integrals, as there are an infinite number of antiderivatives differing by a constant.
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 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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