answer:A) int sin^3(x) , dx = int sin^2(x) sin(x) , dx Using the identity sin^2(x) = 1 - cos^2(x): = int (1 - cos^2(x)) sin(x) , dx Let u = cos(x), then du = -sin(x) , dx: = -int (1 - u^2) , du = -int 1 , du + int u^2 , du = -u + frac{u^3}{3} Substitute back u = cos(x): = -cos(x) + frac{cos^3(x)}{3} Add the constant of integration: int sin^3(x) , dx = -cos(x) + frac{cos^3(x)}{3} + C B) int sin^2(x) cos^2(x) , dx Using the identity sin^2(x) cos^2(x) = frac{1 - cos(4x)}{8}: = int frac{1 - cos(4x)}{8} , dx Take the constant out: = frac{1}{8} int (1 - cos(4x)) , dx Integrate each term: = frac{1}{8} left( int 1 , dx - int cos(4x) , dx right) = frac{1}{8} left( x - frac{sin(4x)}{4} right) Add the constant of integration: int sin^2(x) cos^2(x) , dx = frac{1}{8}x - frac{sin(4x)}{32} + C C) int sin theta cos^5(cos theta) , dtheta Let u = cos(theta), then du = -sin(theta) , dtheta: = -int cos^5(u) , du = -int cos^4(u) cos(u) , du = -int (1 - sin^2(u))^2 cos(u) , du Let v = sin(u), then dv = cos(u) , du: = -int (1 - v^2)^2 , dv = -int (1 - 2v^2 + v^4) , dv Integrate each term: = -left( int 1 , dv - 2int v^2 , dv + int v^4 , dv right) = -left( v - frac{2v^3}{3} + frac{v^5}{5} right) Substitute back u = cos(theta) and v = sin(u): = -left( sin(cos(theta)) - frac{2sin^3(cos(theta))}{3} + frac{sin^5(cos(theta))}{5} right) Add the constant of integration: int sin theta cos^5(cos theta) , dtheta = -sin(cos(theta)) + frac{2sin^3(cos(theta))}{3} - frac{sin^5(cos(theta))}{5} + C

answer:To find the eigenvectors, we first need to determine the eigenvalues. The characteristic equation is given by: [ det(A - lambda I) = 0 ] where (A) is the given matrix, (lambda) is the eigenvalue, and (I) is the identity matrix. For our matrix, this becomes: [ left| begin{array}{cc} -1 - lambda & -frac{15}{4} -frac{15}{2} & frac{25}{4} - lambda end{array} right| = 0 ] Solving this equation, we get: [ lambda^2 - frac{21}{4} lambda - frac{225}{8} = 0 ] Factoring, we find the eigenvalues: [ lambda = frac{1}{2} left(frac{29}{2} pm sqrt{left(frac{29}{2}right)^2 + 4 cdot frac{225}{8}}right) ] [ lambda = frac{1}{2} left(frac{29}{2} pm sqrt{frac{841}{4} + frac{225}{2}}right) ] [ lambda = frac{1}{2} left(frac{29}{2} pm sqrt{frac{841}{4} + frac{450}{4}}right) ] [ lambda = frac{1}{2} left(frac{29}{2} pm sqrt{frac{1291}{4}}right) ] [ lambda = frac{1}{2} left(frac{29}{2} pm frac{sqrt{1291}}{2}right) ] For each eigenvalue (lambda), we solve the system: [ (A - lambda I) mathbf{v} = 0 ] where (mathbf{v}) is the eigenvector. For (lambda_1 = frac{1}{2} left(frac{29}{2} - frac{sqrt{1291}}{2}right)): [ left( begin{array}{cc} -1 - frac{29}{4} + frac{sqrt{1291}}{4} & -frac{15}{4} -frac{15}{2} & frac{25}{4} - frac{29}{4} + frac{sqrt{1291}}{4} end{array} right) mathbf{v_1} = mathbf{0} ] For (lambda_2 = frac{1}{2} left(frac{29}{2} + frac{sqrt{1291}}{2}right)): [ left( begin{array}{cc} -1 - frac{29}{4} - frac{sqrt{1291}}{4} & -frac{15}{4} -frac{15}{2} & frac{25}{4} - frac{29}{4} - frac{sqrt{1291}}{4} end{array} right) mathbf{v_2} = mathbf{0} ] Solving these systems, we find the eigenvectors to be: [ mathbf{v_1} = left{frac{1}{60} left(29-sqrt{1291}right),1right} ] [ mathbf{v_2} = left{frac{1}{60} left(29+sqrt{1291}right),1right} ] Hence, the eigenvectors of the given matrix are: [ mathbf{v_1} = left{frac{1}{60} left(29-sqrt{1291}right),1right}, mathbf{v_2} = left{frac{1}{60} left(29+sqrt{1291}right),1right} ]

answer:This problem can be solved using the combination formula. The number of combinations of n elements taken r at a time is given by: binom{n}{r} = frac{n!}{r!(n-r)!} In this case, we want to choose 4 elements from a set of 7, so the number of combinations is: binom{7}{4} = frac{7!}{4!(7-4)!} = frac{7!}{4!3!} = boxed{35}

answer:To eliminate the parameter ( t ), we set ( x(t) ) equal to ( x ) and solve for ( t ) in terms of ( x ): [ x = frac{400t^2}{9} + 200t + 225 ] Multiplying both sides by 9 to clear the fraction: [ 9x = 400t^2 + 1800t + 2025 ] Rearrange the equation to form a quadratic: [ 400t^2 + 1800t + (2025 - 9x) = 0 ] Divide by 400 to simplify: [ t^2 + frac{9}{4}t + frac{2025 - 9x}{400} = 0 ] Using the quadratic formula: [ t = frac{-frac{9}{4} pm sqrt{left(frac{9}{4}right)^2 - 4 cdot 1 cdot frac{2025 - 9x}{400}}}{2 cdot 1} ] Simplify and solve for ( t ): [ t = frac{-9 pm sqrt{81 - (2025 - 9x)}}{8} ] [ t = frac{-9 pm sqrt{9x - 1944}}{8} ] Now, substitute this expression for ( t ) into ( y(t) ) to find ( y ) in terms of ( x ): [ y = -frac{11}{27}(800left(frac{-9 pm sqrt{9x - 1944}}{8}right)^2 + 3600left(frac{-9 pm sqrt{9x - 1944}}{8}right) + 4041) ] Since both branches of the solution give the same function for ( y ), we can combine them: [ y = -frac{11}{27}(800left(frac{-9 + sqrt{9x - 1944}}{8}right)^2 + 800left(frac{-9 - sqrt{9x - 1944}}{8}right)^2 + 3600left(frac{-9 + sqrt{9x - 1944}}{8}right) + 3600left(frac{-9 - sqrt{9x - 1944}}{8}right) + 4041) ] After simplifying and combining like terms, we get: [ y = frac{11}{3} - frac{22x}{3} ] So, the function ( y = f(x) ) is: [ y = frac{11}{3} - frac{22x}{3} ]