Example Question - solving polynomial integral
Here are examples of questions we've helped users solve.
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.
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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