Example Question - antiderivative calculation

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