Example Question - power rule of integration
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Integration using Power Rule
The image shows an integral expression that you'd like to evaluate. The integral is: ∫ (8x^3 - x^2 + 5x - 1) dx To solve the integral, we use the power rule of integration which states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where n is a real number different from -1, and C is the constant of integration. Let's integrate each term individually: ∫ 8x^3 dx = (8x^(3+1))/(3+1) = (8x^4)/4 = 2x^4 ∫ -x^2 dx = -(x^(2+1))/(2+1) = -(x^3)/3 ∫ 5x dx = (5x^(1+1))/(1+1) = (5x^2)/2 ∫ -1 dx = -1x Putting it all together, the indefinite integral is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Where C is the constant of integration.
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: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \]
Solving a Polynomial Integral
The image contains an integral that you would like to solve. The integral is: ∫(8x^3 - x^2 + 5x - 1) dx To solve this integral, we integrate each term separately with respect to x: ∫8x^3 dx - ∫x^2 dx + ∫5x dx - ∫1 dx Now, for each term, apply the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C where C is the constant of integration: 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, put them all together, including the different signs: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C So the final answer for the indefinite integral is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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