Example Question - power rule integration
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 an Integral using Power Rule for Integration
The image depicts an integral that needs to be solved. The integral is: ∫(8x^3 - x^2 + 5x - 1) dx To solve this integral, you can integrate each term separately. The integral of a sum/difference is the sum/difference of the integrals, and the power rule for integration states that ∫x^n dx = x^(n+1)/(n+1) + C, where C is the constant of integration. Applying this rule to each term gives us: ∫8x^3 dx - ∫x^2 dx + ∫5x dx - ∫1 dx Now integrate each term: 8 * ∫x^3 dx - ∫x^2 dx + 5 * ∫x dx - ∫1 dx = 8 * (x^4/4) - (x^3/3) + 5 * (x^2/2) - x + C = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C This is the antiderivative of the given function. So the solution to the integral is: 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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