Example Question - integration of polynomial functions
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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 Polynomial Functions
The image contains a mathematical expression that represents an integral problem to be solved. Here is the integral from the image: \[ \int (8x^3 - x^2 + 5x - 1) \, dx \] To solve this integral, we will integrate each term separately using the power rule of integration, which states that for any term ax^n, the integral is (a/(n+1))x^(n+1), provided n is not equal to -1. Here are the steps: 1. Integrate \(8x^3\): \[ \int 8x^3 \, dx = \frac{8}{4}x^{3+1} = 2x^4 \] 2. Integrate \(-x^2\): \[ \int -x^2 \, dx = \frac{-1}{3}x^{2+1} = -\frac{1}{3}x^3 \] 3. Integrate \(5x\): \[ \int 5x \, dx = \frac{5}{2}x^{1+1} = \frac{5}{2}x^2 \] 4. Integrate \(-1\): \[ \int -1 \, dx = -x \] Now, let's put all these integrated terms together: \[ \int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] Where \(C\) is the constant of integration, which appears because we are performing indefinite integration. So, the final answer is: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
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