This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to Question 2: Question 2 (a) a) To evaluate x x \, dx, we use integration by parts, u \, dv = uv - v \, du. Step 1: Choose u = x and dv = x \, dx. Step 2: Differentiate u and integrate dv: du = (1)/(x) \, dx v = x \, dx = (x^2)/(2) Step 3: Apply the integration by parts formula: x x \, dx = ( x)((x^2)/(2)) - ((x^2)/(2))((1)/(x)) \, dx Step 4: Simplify and integrate the remaining term: = (x^2)/(2) x - (x)/(2) \, dx = (x^2)/(2) x - (1)/(2) ((x^2)/(2)) + C (x^2)/(2) x - (x^2)/(4) + C b) To evaluate x x \, dx, we use integration by parts. Step 1: Choose u = x and dv = x \, dx. Step 2: Differentiate u and integrate dv: du = dx v = x \, dx = - x Step 3: Apply the integration by parts formula: x x \, dx = (x)(- x) - (- x) \, dx Step 4: Simplify and integrate the remaining term: = -x x + x \, dx -x x + x + C c) To evaluate x^2 e^3x \, dx, we use integration by parts twice. Step 1: For the first application, choose u = x^2 and dv = e^3x \, dx. Step 2: Differentiate u and integrate dv: du = 2x \, dx v = e^3x \, dx = (1)/(3)e^3x Step 3: Apply the integration by parts formula: x^2 e^3x \, dx = x^2 ((1)/(3)e^3x) - ((1)/(3)e^3x)(2x) \, dx = (1)/(3)x^2 e^3x - (2)/(3) x e^3x \, dx Step 4: Now, evaluate x e^3x \, dx using integration by parts again. Choose u = x and dv = e^3x \, dx. Step 5: Differentiate u and integrate dv: du = dx v = e^3x \, dx = (1)/(3)e^3x Step 6: Apply the integration by parts formula for the second time: x e^3x \, dx = x ((1)/(3)e^3x) - ((1)/(3)e^3x) \, dx = (1)/(3)x e^3x - (1)/(3) ((1)/(3)e^3x) + C_1 = (1)/(3)x e^3x - (1)/(9)e^3x + C_1 Step 7: Substitute this result back into the expression from Step 3: x^2 e^3x \, dx = (1)/(3)x^2 e^3x - (2)/(3) ( (1)/(3)x e^3x - (1)/(9)e^3x ) + C (1)/(3)x^2 e^3x - (2)/(9)x e^3x + (2)/(27)e^3x + C Question 2 (b) a) To evaluate (x+1)/(x^2 - 3x + 2) \, dx, we use partial fraction decomposition. Step 1: Factor the denominator: x^2 - 3x + 2 = (x-1)(x-2). Step 2: Set up the partial fraction decomposition: (x+1)/((x-1)(x-2)) = (A)/(x-1) + (B)/(x-2) Multiply both sides by (x-1)(x-2): x+1 = A(x-2) + B(x-1) Step 3: Solve for A and B. Set x=1: 1+1 = A(1-2) 2 = -A A = -2. Set x=2: 2+1 = B(2-1) 3 = B B = 3. Step 4: Rewrite the integral using the partial fractions and integrate: ( (-2)/(x-1) + (3)/(x-2) ) \, dx = -2 (1)/(x-1) \, dx + 3 (1)/(x-2) \, dx -2 |x-1| + 3 |x-2| + C b) To evaluate (x^2+1)/((x+2)^2) \, dx, we can manipulate the numerator. Step 1: Let y = x+2, so x = y-2. Substitute this into the numerator: x^2+1 = (y-2)^2+1 = (y^2 - 4y + 4) + 1 = y^2 - 4y + 5 Step 2: Substitute y back into the integral. Note that dx = dy: (y^2 - 4y + 5)/(y^2) \, dy Step 3: Split the fraction into simpler terms: = ( (y^2)/(y^2) - (4y)/(y^2) + (5)/(y^2) ) \, dy = ( 1 - (4)/(y) + 5y^-2 ) \, dy Step 4: Integrate each term with respect to y: = y - 4 |y| + 5 (y^-1-1) + C = y - 4 |y| - (5)/(y) + C Step 5: Substitute back y = x+2: x+2 - 4 |x+2| - (5)/(x+2) + C That's 2 down. 3 left today — send the next one.