Question - Integration of a Polynomial Function
mathintegration of polynomial functionpolynomial function integrationintegral of polynomial equationpolynomial integral formula
Solution:
Sure, the given integral is:∫ (8x^3 - x^2 + 5x - 1) dxWe need to integrate each term separately. The integral of a polynomial function is found by increasing the exponent by one and dividing by the new exponent. Here's how it works for each term:1. Integral of 8x^3 dx: Increase exponent by 1 (from 3 to 4), then divide by the new exponent (4). ∫ 8x^3 dx = 8/4 x^4 = 2x^42. Integral of -x^2 dx: Increase exponent by 1 (from 2 to 3), then divide by the new exponent (3). ∫ -x^2 dx = -1/3 x^33. Integral of 5x dx: Increase exponent by 1 (from 1 to 2), then divide by the new exponent (2). ∫ 5x dx = 5/2 x^24. Integral of -1 dx: Since the exponent is 0 (because -1 is the same as -1x^0), we just multiply x by the constant. ∫ -1 dx = -1 * x = -xNow, let's put it all together:∫ (8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + CDon't forget to add the constant of integration, C, at the end. The final answer is:2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
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