i}{3}\\right) = \\frac{\\sqrt{3}}{2}$\n* $\\sin(0) = 0$\n\nSubstitute these values back into the expression:\n\n$$\\left( -\\frac{4\\pi}{3} \\cdot \\frac{1}{2} + 4 \\cdot \\frac{\\sqrt{3}}{2} \\right) - (0)$$\n\nSimplify each term:\n\n$$-\\frac{2\\pi}{3} + 2\\sqrt{3}$$"},{"type":"text","order":5,"title":"State Final Exact Answer","content":"Rearranging the terms for a cleaner presentation gives our final exact value:\n\n$$2\\sqrt{3} - \\frac{2\\pi}{3}$$\n\n### Markscheme Breakdown\n* **M1**: Correctly identifying the parts for differentiation ($4x$) and integration ($\\sin x$).\n* **M1**: Applying the integration by parts relation (which is automated by our DI table diagonals).\n* **A1**: Finding the correct antiderivative $[-4x \\cos x + 4 \\sin x]$.\n* **M1**: Substituting the limits of integration $0$ and $\\frac{\\pi}{3}$.\n* **A1**: Correct substitution of the trigonometric values.\n* **A1**: Simplifying to the final exact answer $2\\sqrt{3} - \\frac{2\\pi}{3}$."}]}],"generatedAt":"2026-06-11T03:48:59.294Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"0c24a866-78aa-4003-a555-7ff1ec326c79","revisionId":"100c8647-e639-466a-be83-d6677a2216ed","questionNumber":"2","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"11.9 The Trapezium Rule","calculatorUse":null,"parts":[{"id":"cff6bfc2-20a7-4acc-a45b-e6ceab3a8291","label":"a","content":"Use the trapezium rule with 6 strips to estimate the value of\n\n$$\\int_0^{\\pi} e^x \\sqrt{\\sin x} \\, dx$$","markscheme":"- Calculate step size $h = \\frac{\\pi - 0}{6} = \\frac{\\pi}{6}$ M1\n- Evaluate function at $x = 0, \\frac{\\pi}{6}, \\frac{\\pi}{3}, \\frac{\\pi}{2}, \\frac{2\\pi}{3}, \\frac{5\\pi}{6}, \\pi$ M1\n- Values: $0, 1.1937, 2.6519, 4.8105, 7.5570, 9.6932, 0$ M1\n- Apply formula: $\\frac{\\pi}{6} (1.1937 + 2.6519 + 4.8105 + 7.5570 + 9.6932)$ M2\n- Final answer $13.6$ (3sf) A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Correctly identifies the interval width $h = \\frac{\\pi}{6}$.","code":"M"},{"id":"p2","marks":2,"description":"Correctly evaluates the function at the required $x$ values.","code":"M"},{"id":"p3","marks":2,"description":"Correct substitution into the trapezium rule formula.","code":"M"},{"id":"p4","marks":1,"description":"Final answer of $13.6$ (3sf).","code":"A"}],"finalAnswers":["13.6"],"generatedAt":"2026-05-06T03:41:08.106Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"trapezium_rule","label":"Trapezium Rule","steps":[{"type":"text","order":0,"title":"Calculate step size","content":"To approximate the integral using the trapezium rule with $n = 6$ strips over the interval $[0, \\pi]$, we first determine the width of each strip (the step size, $h$).\n\nThe formula for the step size is:\n$$h = \\frac{b - a}{n}$$\n\nSubstituting our values $a = 0$, $b = \\pi$, and $n = 6$:\n$$h = \\frac{\\pi - 0}{6} = \\frac{\\pi}{6}$$\n\nThis earns the first method mark (**M1**)."},{"type":"text","order":1,"title":"Evaluate the integrand","content":"$40","calculator":[{"gdc":{"expression":"e^x * sqrt(sin(x))","instructions":"1. Set angle mode to Radians.\n2. In Graph or Table mode, enter the function f(x) = e^x * sqrt(sin(x)).\n3. Generate a table from x = 0 to x = π with step π/6."},"ti84":{"expression":"e^(X)*√(sin(X))","instructions":"1. Press [MODE] and ensure 'RADIAN' is highlighted.\n2. Press [Y=] and enter Y1 = e^(X)*√(sin(X)).\n3. Press [2nd] [WINDOW] (TBLSET) and set TblStart = 0 and ΔTbl = π/6.\n4. Press [2nd] [GRAPH] (TABLE) to see the table of values."},"desmos":{"expression":"f(x) = e^x \\sqrt{\\sin(x)}","instructions":"1. Set the graph to Radian mode.\n2. Create the function f(x) = e^x * sqrt(sin(x)).\n3. Add a table and enter x-values: 0, π/6, π/3, π/2, 2π/3, 5π/6, π to get the corresponding y-values."},"description":"Calculate function values at specific $x$ coordinates usin

Integration

Question 43

Integration Questions

Integration

Question 43

Integration Questions