Question - Solving an Integral using Power Rule for Integration
Solution:
The image depicts an integral that needs to be solved. The integral is:∫(8x^3 - x^2 + 5x - 1) dxTo 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 dxNow 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 + CThis 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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