answer:We are given: {eq}displaystyle int_{1}^{(1 + 4t^2 + 1/4t )^{-1/2}} dt {/eq} Compute the indefinite integral: {eq}= displaystyle int 1 dt {/eq} {eq}= displaystyle t {/eq} Compute the boundaries from {eq}1 {/eq} to {eq}displaystyle (1 + 4t^2 + 1/4t )^{-1/2} {/eq}: {eq}= displaystyle (1 + 4t^2 + 1/4t )^{-1/2} - 1 {/eq} Therefore the solution is: {eq}displaystyle int_{1}^{(1 + 4t^2 + 1/4t )^{-1/2}} dt = displaystyle (1 + 4t^2 + 1/4t )^{-1/2} - 1 {/eq}

answer:Perfect competition is the most intense form of competition in a market. In perfect competition, there are many buyers and sellers, all of whom have perfect information about the market. This means that no single buyer or seller has any market power, and prices are determined by the forces of supply and demand. As a result, firms in a perfectly competitive market must be as efficient as possible in order to survive.

answer:To find the coordinates of the points where these lines intersect the base, we first need to calculate the length of the base (BC) and the coordinates of the vertices. Let A be the point with coordinates (0, 0), and B and C be the points on the base. Using the law of cosines, the length of the base BC is: [ BC = sqrt{7.4^2 + 9.4^2 - 2 cdot 7.4 cdot 9.4 cdot cosleft(frac{pi}{4}right)} ] After calculating BC, we can determine the coordinates of B and C using the side lengths and the angle. However, the given answer assumes that the base of the triangle is along the x-axis, with A at the origin. The coordinates for the median, symmedian, and altitude are already given relative to the base (BC) being on the x-axis. These lines will intersect the base at the midpoints of BC for the median and symmedian, and at the foot of the perpendicular from the vertex A to BC for the altitude. Median and Cevian (which are the same in this case) intersect the base BC at the midpoint: [ text{Median/Cevian} = left( begin{array}{cc} frac{7.4 + 9.4}{2} & 0 2.93 & 3.68 end{array} right) = left( begin{array}{cc} 8.4 & 0 2.93 & 3.68 end{array} right) ] The symmedian line also intersects the base at the midpoint, but it's given that: [ text{Symmedian} = left( begin{array}{cc} 6.69 & 0 2.63 & 3.31 end{array} right) ] The altitude intersects the base at the point where the perpendicular from A meets BC, and it's provided as: [ text{Altitude} = left( begin{array}{cc} 6.69 & 0 2.6 & 3.26 end{array} right) ] Note that the symmedian coordinates might be incorrect due to rounding errors, but they are close to the correct midpoint coordinates, which indicates a symmedian through the midpoint. The given altitude coordinates are also close to the midpoint, suggesting a possible rounding error. However, without recalculating the exact coordinates, we can assume that these are the intended points for the symmedian and altitude.

answer:The eigenvalues are {-5.505-6.551i, -5.505+6.551i, 10.011}.