Question - Integration of Polynomial Functions
mathindefinite integrationintegration of polynomial functionspower rule of integrationintegral problem solving
Solution:
The image contains a mathematical expression that represents an integral problem to be solved. Here is the integral from the image:$$\int (8x^3 - x^2 + 5x - 1) \, dx$$To solve this integral, we will integrate each term separately using the power rule of integration, which states that for any term ax^n, the integral is (a/(n+1))x^(n+1), provided n is not equal to -1.Here are the steps:1. Integrate $$8x^3$$: $$ \int 8x^3 \, dx = \frac{8}{4}x^{3+1} = 2x^4 \]2. Integrate $$-x^2$$: $$ \int -x^2 \, dx = \frac{-1}{3}x^{2+1} = -\frac{1}{3}x^3 \]3. Integrate $$5x$$: $$ \int 5x \, dx = \frac{5}{2}x^{1+1} = \frac{5}{2}x^2 \]4. Integrate $$-1$$: $$ \int -1 \, dx = -x \]Now, let's put all these integrated terms together:$$\int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C$$Where $$C$$ is the constant of integration, which appears because we are performing indefinite integration.So, the final answer is:$$2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C$$
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