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Problem 6) a) Express f(x) in partial fractions where f(x) = (2)/((x-1)(x+1)). (Assuming the 'b' in the denominator is a typo and the function is f(x) = (2)/((x-1)(x+1)) to align with the subsequent integral.) Step 1: Set up the partial fraction decomposition. We want to find constants A and B such that: (2)/((x-1)(x+1)) = (A)/(x-1) + (B)/(x+1) Step 2: Combine the terms on the right-hand side. (2)/((x-1)(x+1)) = (A(x+1) + B(x-1))/((x-1)(x+1)) Equating the numerators: 2 = A(x+1) + B(x-1) Step 3: Solve for A and B. To find A, let x=1: 2 = A(1+1) + B(1-1) 2 = 2A + 0 A = 1 To find B, let x=-1: 2 = A(-1+1) + B(-1-1) 2 = 0 - 2B B = -1 Step 4: Write the partial fraction decomposition. f(x) = (1)/(x-1) - (1)/(x+1) b) Hence using the substitution x=t^2-2, or otherwise, show that _2^7 (1)/((x+1)sqrt(x+2)) dx = ((3)/(2)). Step 1: Apply the substitution x=t^2-2. If x = t^2-2, then dx = 2t \, dt. Step 2: Change the limits of integration. When x=2: 2 = t^2 - 2 t^2 = 4 t = 2 (since t 0 for sqrt(x+2)) When x=7: 7 = t^2 - 2 t^2 = 9 t = 3 (since t 0) Step 3: Substitute into the integral. The term sqrt(x+2) becomes: sqrt(x+2) = sqrt((t^2-2)+2) = sqrt(t^2) = t The term (x+1) becomes: x+1 = (t^2-2)+1 = t^2-1 Substitute these into the integral: _2^7 (1)/((x+1)sqrt(x+2)) dx = _2^3 (1)/((t^2-1)t) (2t \, dt) = _2^3 (2t)/((t^2-1)t) dt = _2^3 (2)/(t^2-1) dt = _2^3 (2)/((t-1)(t+1)) dt Step 4: Use the partial fraction decomposition from part (a). From part (a), we know that (2)/((t-1)(t+1)) = (1)/(t-1) - (1)/(t+1). So the integral becomes: _2^3 ((1)/(t-1) - (1)/(t+1)) dt Step 5: Integrate the partial fractions. [|t-1| - |t+1|]_2^3 Using the logarithm property a - b = ((a)/(b)): [|(t-1)/(t+1)|]_2^3 Step 6: Evaluate the definite integral. Substitute the upper limit t=3: |(3-1)/(3+1)| = |(2)/(4)| = ((1)/(2)) Substitute the lower limit t=2: |(2-1)/(2+1)| = |(1)/(3)| = ((1)/(3)) Subtract the lower limit value from the upper limit value: ((1)/(2)) - ((1)/(3)) = ((1/2)/(1/3)) = ((1)/(2) × (3)/(1)) = ((3)/(2)) Thus, we have shown that _2^7 (1)/((x+1)sqrt(x+2)) dx = ((3)/(2)). What's next?