Question - Antiderivative of a Polynomial Function
mathconstant of integrationantiderivative of polynomial functionintegral of polynomial functionpower rule for integration
Solution:
Certainly! The expression you provided is an integral, and we will find the antiderivative (indefinite integral) of the function. Here's the integral:$$\int (8x^3 - x^2 + 5x - 1) \, dx$$To solve this integral, we will integrate each term separately. Recall the power rule for integration, $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where C is the constant of integration. Applying this rule to each term:1. The integral of $$8x^3$$ is $$\frac{8}{4}x^{4} = 2x^4$$.2. The integral of $$-x^2$$ is $$-\frac{1}{3}x^{3} = -\frac{1}{3}x^3$$.3. The integral of $$5x$$ is $$\frac{5}{2}x^{2} = \frac{5}{2}x^2$$.4. The integral of $$-1$$ is $$-x$$.Now, putting it all together and adding the constant of integration, $$C$$, the antiderivative is:$$\int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C$$
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