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Integration of (2x^3 + x^2) dx

Answer:

Integration of (2x^3 + x^2) dx is the process of finding the anti-derivative of the expression. The anti-derivative of 2x^3 + x^2 is 2x^4/4 + x^3/3 + C, where C is an arbitrary constant. We can use the power rule of integration to break down the expression and integrate it.

Power Rule of Integration

The power rule of integration states that if we have an expression of the form f(x) = ax^n, the anti-derivative of the expression is F(x) = ax^(n+1)/(n+1) + C. This rule is used to integrate any expression containing a power of x, including polynomials.

Integrating (2x^3 + x^2)

We can use the power rule of integration to integrate (2x^3 + x^2). We start by breaking down the expression into its individual terms. The first term is 2x^3, which can be written as 2x^3 = 2x^3/3 = 2/3 * x^3. Applying the power rule of integration to this term yields 2x^4/4 + C. The second term is x^2. Applying the power rule of integration to this term yields x^3/3 + C. Finally, we add the two anti-derivatives together to get the final anti-derivative: 2x^4/4 + x^3/3 + C.

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Suppose 20 households in a community have the demand function Q = 2 – 0.001P each for streetlights while 5 other households have the demand function Q = 3 – 0.002P each for streetlights. Here, Q refers to the number of streetlights to erect near their home while P is the unit price of each streetlight. The cost of building each streetlight is GHS 45. a) Assuming each household is making its separate purchase decision, find the market demand function for streetlights for all 25 households b) Assuming each household is making its separate purchase decision, find the equilibrium price and number of streetlights in the community c) Assuming a community association has decided to coordinate this streetlight purchase decision problem. The community leader uses the demand functions to determine how much each household is willing to contribute (willingness to pay) towards the first, second, third, etc streetlight. The first light is purchased only when the sum of individual households’ contributions is at least equal to the marginal cost of streetlights. Find out how many streetlights are purchased for this community in equilibrium. Explain your answers in details

Suppose 20 households in a community have the demand function Q = 2 – 0.001P each for streetlights while 5 other households have the demand function Q = 3 – 0.002P each for streetlights. Here, Q refers to the number of streetlights to erect near their home while P is the unit price of each streetlight. The cost of building each streetlight is GHS 45. a) Assuming each household is making its separate purchase decision, find the market demand function for streetlights for all 25 households b) Assuming each household is making its separate purchase decision, find the equilibrium price and number of streetlights in the community c) Assuming a community association has decided to coordinate this streetlight purchase decision problem. The community leader uses the demand functions to determine how much each household is willing to contribute (willingness to pay) towards the first, second, third, etc streetlight. The first light is purchased only when the sum of individual households’ contributions is at least equal to the marginal cost of streetlights. Find out how many streetlights are purchased for this community in equilibrium.

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