answer:Democracy is a system of government in which the people have the authority to choose their governing legislation. It is derived from the Greek words "demos," which means "people," and "kratos," which means "power." In a democracy, the citizens have the right to participate in the decision-making process, either directly or through elected representatives. There are various forms of democracy, including direct democracy, where citizens make decisions directly, and representative democracy, where citizens elect representatives to make decisions on their behalf. A form of government in which the people, or some significant portion of them, have supreme control over the government and where offices of state are elected or chosen by elected people.

answer:We will apply integration by parts multiple times to evaluate this integral. The formula for integration by parts is {eq}int u , dv = uv - int v , du {/eq}, where {eq}u {/eq} and {eq}v {/eq} are differentiable functions. First iteration: Let {eq}u = x^2 {/eq} and {eq}dv = e^{5x} , mathrm{d}x {/eq}. Then, {eq}du = 2x , mathrm{d}x {/eq} and {eq}v = frac{1}{5}e^{5x} {/eq}. Applying integration by parts: {eq}int x^2 e^{5x} , mathrm{d}x = frac{x^2}{5}e^{5x} - int frac{2}{5}xe^{5x} , mathrm{d}x {/eq} For the remaining integral, we apply integration by parts again: Let {eq}u = x {/eq} and {eq}dv = frac{2}{5}e^{5x} , mathrm{d}x {/eq}. Then, {eq}du = mathrm{d}x {/eq} and {eq}v = frac{1}{25}e^{5x} {/eq}. {eq}int frac{2}{5}xe^{5x} , mathrm{d}x = frac{2}{5}x cdot frac{1}{25}e^{5x} - int frac{1}{25}e^{5x} , mathrm{d}x = frac{2x}{125}e^{5x} - frac{1}{125}e^{5x} {/eq} Now, substitute this back into the original integral: {eq}int x^2 e^{5x} , mathrm{d}x = frac{x^2}{5}e^{5x} - left(frac{2x}{125}e^{5x} - frac{1}{125}e^{5x}right) {/eq} Simplify the expression: {eq}int x^2 e^{5x} , mathrm{d}x = frac{x^2}{5}e^{5x} - frac{2x}{125}e^{5x} + frac{1}{125}e^{5x} {/eq} This is the final integral in terms of {eq}x {/eq}.

answer:Given the arithmetic sequence with a_1=frac{23}{42} and a common difference of -3sqrt{5}, the nth partial sum S_n can be found using the formula: [ S_n = frac{n}{2} left(2a_1 + (n-1)dright) ] where d is the common difference. In this case, d = -3sqrt{5} and n = 13. Plugging in these values, we get: [ S_{13} = frac{13}{2} left(2 cdot frac{23}{42} + (13-1)(-3sqrt{5})right) ] [ S_{13} = frac{13}{2} left(frac{46}{42} - 36sqrt{5}right) ] [ S_{13} = frac{13}{2} left(frac{23}{21} - 36sqrt{5}right) ] [ S_{13} = frac{13}{2} left(frac{23 - 21 cdot 72sqrt{5}}{21}right) ] [ S_{13} = frac{13}{2} left(frac{23 - 1512sqrt{5}}{21}right) ] [ S_{13} = frac{13(23 - 1512sqrt{5})}{2 cdot 21} ] [ S_{13} = frac{13(23)}{42} - frac{13(1512sqrt{5})}{42} ] [ S_{13} = frac{299}{42} - frac{19656sqrt{5}}{42} ] [ S_{13} = frac{299 - 19656sqrt{5}}{42} ] [ S_{13} = frac{299 - 19656sqrt{5}}{42} ] This is the nth partial sum of the given arithmetic sequence when n=13.

answer:To compute the cube of a matrix, we need to multiply the matrix by itself three times. First, we compute the square of the matrix: left( begin{array}{ccc} -1 & 2 & frac{3}{2} -2 & -frac{5}{2} & frac{1}{2} -frac{1}{2} & 3 & -frac{5}{2} end{array} right)^2 = left( begin{array}{ccc} frac{1}{2} & -frac{13}{2} & frac{1}{2} -frac{13}{2} & -frac{29}{2} & -frac{13}{2} frac{1}{2} & -frac{13}{2} & frac{1}{2} end{array} right). Then, we multiply the square of the matrix by the original matrix: left( begin{array}{ccc} frac{1}{2} & -frac{13}{2} & frac{1}{2} -frac{13}{2} & -frac{29}{2} & -frac{13}{2} frac{1}{2} & -frac{13}{2} & frac{1}{2} end{array} right) times left( begin{array}{ccc} -1 & 2 & frac{3}{2} -2 & -frac{5}{2} & frac{1}{2} -frac{1}{2} & 3 & -frac{5}{2} end{array} right) = left( begin{array}{ccc} frac{87}{8} & -14 & frac{15}{4} -frac{23}{2} & -frac{99}{8} & frac{103}{4} frac{131}{4} & frac{105}{2} & -frac{255}{8} end{array} right). Therefore, the cube of the given matrix is left( begin{array}{ccc} frac{87}{8} & -14 & frac{15}{4} -frac{23}{2} & -frac{99}{8} & frac{103}{4} frac{131}{4} & frac{105}{2} & -frac{255}{8} end{array} right). The answer is left( begin{array}{ccc} frac{87}{8} & -14 & frac{15}{4} -frac{23}{2} & -frac{99}{8} & frac{103}{4} frac{131}{4} & frac{105}{2} & -frac{255}{8} end{array} right).