a) Step 1: Express the rational function in partial fractions.

a) Step 1: Express the rational function in partial fractions. We set up the partial fraction decomposition for (5x-2)/(x^2(x+2)): (5x-2)/(x^2(x+2)) = (A)/(x) + (B)/(x^2) + (C)/(x+2) Multiply both sides by x^2(x+2) to clear the denominators: 5x-2 = A x(x+2) + B(x+2) + C x^2 5x-2 = Ax^2 + 2Ax + Bx + 2B + C x^2 5x-2 = (A+C)x^2 + (2A+B)x + 2B Step 2: Solve for the constants A, B, and C. By comparing coefficients: For the constant term: 2B = -2 B = -1. For the coefficient of x: 2A+B = 5. Substitute B=-1: 2A-1 = 5 2A = 6 A = 3. For the coefficient of x^2: A+C = 0. Substitute A=3: 3+C = 0 C = -3. Step 3: Write the partial fraction decomposition. (5x-2)/(x^2(x+2)) = (3)/(x) - (1)/(x^2) - (3)/(x+2) Step 4: Integrate the partial fractions. (5x-2)/(x^2(x+2))dx = ((3)/(x) - (1)/(x^2) - (3)/(x+2))dx = (3)/(x)dx - x^-2dx - (3)/(x+2)dx = 3 |x| - x^-1-1 - 3 |x+2| + K = 3 |x| + (1)/(x) - 3 |x+2| + K Combine the logarithmic terms: = 3 (|x| - |x+2|) + (1)/(x) + K = 3 |(x)/(x+2)| + (1)/(x) + K The partial fraction decomposition is (3)/(x) - (1)/(x^2) - (3)/(x+2) and the integral is 3 |(x)/(x+2)| + (1)/(x) + K. b) Step 1: Move all terms to one side and combine into a single fraction. (x+1)/(x-1) 2 (x+1)/(x-1) - 2 0 (x+1 - 2(x-1))/(x-1) 0 (x+1 - 2x + 2)/(x-1) 0 (-x+3)/(x-1) 0 Step 2: Find the critical points. The critical points are where the numerator is zero or the denominator is zero. Numerator: -x+3 = 0 x=3. Denominator: x-1 = 0 x=1. Step 3: Test intervals defined by the critical points. The critical points x=1 and x=3 divide the number line into three intervals: (-, 1), (1, 3), and (3, ). For x < 1 (e.g., x=0): (-0+3)/(0-1) = (3)/(-1) = -3. This is not 0. For 1 < x < 3 (e.g., x=2): (-2+3)/(2-1) = (1)/(1) = 1. This is 0. For x > 3 (e.g., x=4): (-4+3)/(4-1) = (-1)/(3). This is not 0. Step 4: Check the endpoints. At x=3: (-3+3)/(3-1) = (0)/(2) = 0. Since 0 0, x=3 is included in the solution. At x=1: The denominator is zero, so the expression is undefined. Thus, x=1 is not included. Step 5: State the solution. The inequality holds for 1 < x 3. The solution is 1 < x 3. c) Step 1: Identify the letters and their frequencies in the word STATISTICS. The word STATISTICS has 10 letters: S: 3 T: 3 A: 1 I: 2 C: 1 Step 2: Treat the sub-word ACT as a single unit. When "ACT" is treated as a single unit, we consider it as one block. The letters used in "ACT" are one A, one C, and one T. Remaining letters from STATISTICS: S: 3 T: 3-1 = 2 A: 1-1 = 0 I: 2 C: 1-1 = 0 Step 3: List the items to be arranged and their frequencies. The items to arrange are the block (ACT) and the remaining letters: (ACT): 1 unit S: 3 times T: 2 times I: 2 times The total number of items to arrange is 1+3+2+2 = 8. Step 4: Calculate the number of distinct arrangements. This is a permutation with repetitions. The formula is $N!n_1