Question - Integrating a Polynomial using Power Rule
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Solution:
You are asked to integrate a polynomial. The integral of a polynomial is found by integrating each term separately, according to the power rule for integration, $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ where $$C$$ is the constant of integration.The integral of the polynomial $$8x^3 - x^2 + 5x - 1$$ is calculated as follows:$$\int (8x^3 - x^2 + 5x - 1)dx = \int 8x^3 dx - \int x^2 dx + \int 5x dx - \int 1dx$$Now integrate each term separately:$$\int 8x^3 dx = 8 \times \frac{x^{3+1}}{3+1} = 2x^4$$$$\int x^2 dx = \frac{x^{2+1}}{2+1} = \frac{1}{3}x^3$$$$\int 5x dx = 5 \times \frac{x^{1+1}}{1+1} = \frac{5}{2}x^2$$$$\int 1dx = x$$So the integral is:$$2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C$$where $$C$$ is the constant of integration.
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