Question - Integration of a Polynomial Function
mathintegral of polynomial functionpower rule for integrationintegrating polynomial equationspolynomial function integration
Solution:
To solve the given integral, we will integrate each term separately:$$\int (8x^3 - x^2 + 5x - 1)\ dx$$We integrate each term using the power rule for integration, which is:$$\int x^n\ dx = \frac{x^{n+1}}{n+1} + C$$where $$n \neq -1$$ and $$C$$ is the constant of integration.Now, integrating each term:1. $$\int 8x^3\ dx = \frac{8}{4}x^{3+1} = 2x^4$$2. $$\int -x^2\ dx = -\frac{1}{3}x^{2+1} = -\frac{1}{3}x^3$$3. $$\int 5x\ dx = \frac{5}{2}x^{1+1} = \frac{5}{2}x^2$$4. $$\int -1\ dx = -x$$Combine these results and add the constant of integration ($$C$$):$$\int (8x^3 - x^2 + 5x - 1)\ dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C$$So, the integral of $$8x^3 - x^2 + 5x - 1$$ with respect to $$x$$ is:$$2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C$$
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