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They allow us to solve integrals involving absolute values, symmetric limits, and periodic functions, often bypassing the need to find the explicit antiderivative first." }) }), _jsx(Hinglish, { children: _jsx(_components.p, { children: "Properties of Definite Integrals aisi rules aur theorems hain jo complex integrations ko short aur aasan tarike se solve karne mein help karti hain. In properties ka use karke hum symmetric limits, modulus (absolute value), aur heavy trigonometric integrals ko bina directly integrate kiye solve kar sakte hain." }) })] }), "\n", _jsxs(Explanation, { children: [_jsxs(English, { children: [_jsx(_components.h3, { children: "Key Properties of Definite Integrals" }), _jsx(_components.p, { children: "Here is the list of crucial properties frequently tested in MSBTE examinations:" }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Change of Variable:" }), "\n", _jsx(_components.span, { className: "katex", children: _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.3998em", verticalAlign: "-0.3558em" } }), _jsxs(_components.span, { className: "mop", children: [_jsx(_components.span, 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})] })] }) })] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.p, { children: _jsx(_components.strong, { children: "Symmetric Limits (Even/Odd Functions):" }) }), "\n", _jsx(_components.span, { className: "katex-error", title: "ParseError: KaTeX parse error: Expected 'EOF', got '&' at position 23: …0}^{a} f(x) dx &̲ \\text{if } f(x…", style: { color: "#cc0000" }, children: "2\\int_{0}^{a} f(x) dx & \\text{if } f(x) \\text{ is an Even function } (f(-x) = f(x)) \\\\ \n0 & \\text{if } f(x) \\text{ is an Odd function } (f(-x) = -f(x)) \n\\end{cases}$$" }), "\n"] }), "\n"] }), _jsx(_components.h3, { children: "Even vs. Odd Functions Examples" })] }), _jsxs(Hinglish, { children: [_jsx(_components.h3, { children: "Definite Integrals Ki Main Properties" }), _jsx(_components.p, { children: "MSBTE exams ke liye important definite integration properties niche di gayi hain:" }), _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["\n", 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children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "a" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.2579em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "b" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.3558em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, 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className: "mop op-symbol small-op", style: { marginRight: "0.1945em", position: "relative", top: "-0.0006em" }, children: "∫" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.044em" }, children: [_jsxs(_components.span, { style: { top: "-2.3442em", marginLeft: "-0.1945em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "a" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.2579em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), 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\\\\ \n0 & \\text{agar } f(x) \\text{ Odd function hai } (f(-x) = -f(x)) \n\\end{cases}$$" }), "\n"] }), "\n"] }), _jsx(_components.h3, { children: "Even aur Odd Functions Ke Examples" })] }), _jsx(ComparisonTable, { children: _jsxs(_components.table, { children: [_jsx(_components.thead, { children: _jsxs(_components.tr, { children: [_jsxs(_components.th, { style: { textAlign: "left" }, children: ["Function ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.1076em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mclose", 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}), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mop", children: "cos" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "x" })] })] }) }) }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.strong, { children: "Odd" }) }), _jsx(_components.td, { style: { textAlign: "left" }, children: _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] }) }) }) })] })] })] }) })] }), "\n", _jsxs(Summary, { children: [_jsx(English, { children: _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Odd functions evaluated over symmetric limits ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) }) }), " always yield zero."] }), "\n", _jsx(_components.li, { children: "Modulus functions must be split into sections where the term inside is positive or negative." }), "\n", _jsx(_components.li, { children: "King's property is primarily used to solve trigonometric fractions by adding the original and modified equations." }), "\n"] }) }), _jsx(Hinglish, { children: _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: ["Symmetric limits ", _jsx(_components.span, { className: "katex", children: _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord", children: "−" }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mpunct", children: "," }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mclose", children: "]" })] }) }) }), " ke limits mein Odd functions ka integration hamesha zero hota hai."] }), "\n", _jsx(_components.li, { children: "Modulus functions ko split point par break karke do alag sections mein integrate karte hain (positive aur negative intervals ke liye)." }), "\n", _jsx(_components.li, { children: "Trigonometric fractions ko solve karne ke liye King's property lagakar dono forms ko add kiya jata hai jisse terms cut jate hain." }), "\n"] }) })] }), "\n", _jsx(ImportantQA, { questions: [{ "q": "Evaluate: Integral of sin^3(x) dx from -pi/2 to pi/2.", "marks": 2, "a": { "components": [{ "type": "Explanation", "content": "Given: $I = \\int_{-\\pi/2}^{\\pi/2} \\sin^3 x dx$.\nLet $f(x) = \\sin^3 x$.\nCheck if the function is even or odd:\n$f(-x) = [\\sin(-x)]^3 = [-\\sin x]^3 = -\\sin^3 x = -f(x)$.\nTherefore, $f(x)$ is an odd function.\nBy the symmetric limits property, since the function is odd:\n$I = 0$.Diya hai: $I = \\int_{-\\pi/2}^{\\pi/2} \\sin^3 x dx$.\nLet $f(x) = \\sin^3 x$.\nEven/Odd check karne par:\n$f(-x) = [\\sin(-x)]^3 = [-\\sin x]^3 = -\\sin^3 x = -f(x)$.\nIsliye, $f(x)$ ek odd function hai.\nSymmetric limits property ke mutabik odd function ka integral:\n$I = 0$." }] } }, { "q": "Evaluate: Integral of sqrt(sin(x)) / (sqrt(sin(x)) + sqrt(cos(x))) dx from 0 to pi/2.", "marks": 4, "a": { "components": [{ "type": "Explanation", "content": "Let $I = \\int_{0}^{\\pi/2} \\frac{\\sqrt{\\sin x}}{\\sqrt{\\sin x} + \\sqrt{\\cos x}} dx$ --- (Eq. 1).\nUsing the property $\\int_{0}^{a} f(x) dx = \\int_{0}^{a} f(a - x) dx$:\n$I = \\int_{0}^{\\pi/2} \\frac{\\sqrt{\\sin(\\pi/2 - x)}}{\\sqrt{\\sin(\\pi/2 - x)} + \\sqrt{\\cos(\\pi/2 - x)}} dx$\nSince $\\sin(\\pi/2 - x) = \\cos x$ and $\\cos(\\pi/2 - x) = \\sin x$:\n$I = \\int_{0}^{\\pi/2} \\frac{\\sqrt{\\cos x}}{\\sqrt{\\cos x} + \\sqrt{\\sin x}} dx$ --- (Eq. 2).\nAdding Eq. 1 and Eq. 2:\n$2I = \\int_{0}^{\\pi/2} \\frac{\\sqrt{\\sin x} + \\sqrt{\\cos x}}{\\sqrt{\\sin x} + \\sqrt{\\cos x}} dx$\n$2I = \\int_{0}^{\\pi/2} 1 dx$\n$2I = [x]_{0}^{\\pi/2} = \\pi/2$\n$I = \\pi/4$.Let $I = \\int_{0}^{\\pi/2} \\frac{\\sqrt{\\sin x}}{\\sqrt{\\sin x} + \\sqrt{\\cos x}} dx$ --- (Eq. 1).\nProperty $\\int_{0}^{a} f(x) dx = \\int_{0}^{a} f(a - x) dx$ use karne par:\n$I = \\int_{0}^{\\pi/2} \\frac{\\sqrt{\\sin(\\pi/2 - x)}}{\\sqrt{\\sin(\\pi/2 - x)} + \\sqrt{\\cos(\\pi/2 - x)}} dx$\nKyunki $\\sin(\\pi/2 - x) = \\cos x$ aur $\\cos(\\pi/2 - x) = \\sin x$:\n$I = \\int_{0}^{\\pi/2} \\frac{\\sqrt{\\cos x}}{\\sqrt{\\cos x} + \\sqrt{\\sin x}} dx$ --- (Eq. 2).\nEq. 1 aur Eq. 2 ko add karne par:\n$2I = \\int_{0}^{\\pi/2} \\frac{\\sqrt{\\sin x} + \\sqrt{\\cos x}}{\\sqrt{\\sin x} + \\sqrt{\\cos x}} dx$\n$2I = \\int_{0}^{\\pi/2} 1 dx$\n$2I = [x]_{0}^{\\pi/2} = \\pi/2$\n$I = \\pi/4$." }] } }, { "q": "Evaluate: Integral of |x - 2| dx from 1 to 3.", "marks": 4, "a": { "components": [{ "type": "Explanation", "content": "Given: $I = \\int_{1}^{3} |x - 2| dx$.\nThe modulus function changes definition at $x = 2$:\n- For $x < 2$, $|x - 2| = -(x - 2) = 2 - x$\n- For $x \\ge 2$, $|x - 2| = x - 2$\nSplitting the integral at $x = 2$ using properties:\n$I = \\int_{1}^{2} (2 - x) dx + \\int_{2}^{3} (x - 2) dx$\n$= \\left[ 2x - \\frac{x^2}{2} \\right]_{1}^{2} + \\left[ \\frac{x^2}{2} - 2x \\right]_{2}^{3}$\nFirst part: $\\left(4 - 2\\right) - \\left(2 - \\frac{1}{2}\\right) = 2 - 1.5 = 0.5$.\nSecond part: $\\left(\\frac{9}{2} - 6\\right) - \\left(2 - 4\\right) = -1.5 - (-2) = 0.5$.\nTotal integral $I = 0.5 + 0.5 = 1$.Diya hai: $I = \\int_{1}^{3} |x - 2| dx$.\nModulus function $x = 2$ par value badalta hai:\n- For $x < 2$, $|x - 2| = 2 - x$\n- For $x \\ge 2$, $|x - 2| = x - 2$\nProperty se integral ko $x = 2$ par split karne par:\n$I = \\int_{1}^{2} (2 - x) dx + \\int_{2}^{3} (x - 2) dx$\n$= \\left[ 2x - \\frac{x^2}{2} \\right]_{1}^{2} + \\left[ \\frac{x^2}{2} - 2x \\right]_{2}^{3}$\nFirst part: $\\left(4 - 2\\right) - \\left(2 - 1/2\\right) = 2 - 1.5 = 0.5$.\nSecond part: $\\left(9/2 - 6\\right) - \\left(2 - 4\\right) = -1.5 - (-2) = 0.5$.\nTotal $I = 0.5 + 0.5 = 1$." }] } }] }), "\n", _jsx(MCQ, { questions: [{ "id": "5d1541c9-c4c9-45c0-88fb-0376466393c7", "q": "What is the value of Integral of x^3 * cos(x) dx from -pi to pi?", "options": ["pi", "0", "2 * pi", "1"], "answer": 1, "hint": "Check if f(x) = x^3 * cos(x) is even or odd.", "explanation": "f(-x) = (-x)^3 * cos(-x) = -x^3 * cos(x) = -f(x). Since f(x) is an odd function and limits are symmetric, the integral is 0." }, { "id": "f989a542-60f2-48ea-ab20-c372b47bd36c", "q": "Which of the following describes an even function f(x)?", "options": ["f(-x) = -f(x)", "f(-x) = f(x)", "f(x) = 0", "f'(x) = f(x)"], "answer": 1, "hint": "Even functions ignore the negative sign (like x^2 or cos(x)).", "explanation": "An even function satisfies the property f(-x) = f(x) for all x in its domain." }, { "id": "9a7b3767-1ef6-4f9e-ac4c-b876e1746ff3", "q": "What is the value of the integral of |x| dx from -2 to 2?", "options": ["0", "2", "4", "8"], "answer": 2, "hint": "f(x) = |x| is an even function, so the integral is 2 * (Integral of x dx from 0 to 2).", "explanation": "Since |x| is even, the integral is 2 * Integral[0, 2] x dx = 2 * [x^2 / 2] from 0 to 2 = 2 * (4/2 - 0) = 4." }, { "id": "2c9225b8-d819-4fa7-8893-5c66c1e2d7f4", "q": "Using the properties of definite integrals, evaluate the integral of log(tan(x)) dx from 0 to pi/2.", "options": ["0", "pi/2", "log(2)", "pi/4"], "answer": 0, "hint": "Apply the King's Property: tan(pi/2 - x) = cot(x) = 1/tan(x). Add both equations.", "explanation": "Let I = Integral log(tan x). Using property, I = Integral log(cot x) = Integral log(1/tan x) = -I. Thus, 2I = 0 => I = 0." }, { "id": "1f6caeb9-9a53-47ea-b311-461bdd13c162", "q": "Which of the following functions is an ODD function?", "options": ["x^2 + 1", "cos(x)", "x^5 + sin(x)", "e^x"], "answer": 2, "hint": "Check which function yields -f(x) when x is replaced by -x.", "explanation": "For f(x) = x^5 + sin(x), f(-x) = (-x)^5 + sin(-x) = -x^5 - sin(x) = -f(x), making it an odd function." }, { "id": "8c3ca55e-a4af-401b-be28-c234ed47cf40", "q": "Evaluate: Integral of x^4 dx from -3 to 3.", "options": ["0", "2 * (Integral of x^4 dx from 0 to 3)", "2 * 3^4", "81"], "answer": 1, "hint": "f(x) = x^4 is an even function. Apply symmetric limits property.", "explanation": "Since x^4 is even, the integral over symmetric limits [-3, 3] becomes twice the integral from 0 to 3." }, { "id": "c0cf3127-ae76-496b-a041-b1c3ba19d17b", "q": "Which property allows us to write Integral of f(x) dx from 0 to 10 as Integral of f(x) dx from 0 to 4 plus Integral from 4 to 10?", "options": ["Limits reversing property", "Splitting the interval property", "King's property", "Even-Odd function property"], "answer": 1, "hint": "The interval [0, 10] is split into [0, 4] and [4, 10].", "explanation": "The splitting property states that an integral can be split at any point c within the interval [a, b]." }, { "id": "a32ba3ef-0475-4791-ad38-428e2d2e5b3f", "q": "Evaluate the definite integral of cos(x)/(sin(x) + cos(x)) dx from 0 to pi/2.", "options": ["pi/2", "pi/4", "0", "pi"], "answer": 1, "hint": "Use the property f(x) = f(pi/2 - x) and add the two equations to get 2I = Integral of 1 dx.", "explanation": "Applying the property gives I = Integral sin(x)/(cos(x) + sin(x)). Adding them gives 2I = Integral[0, pi/2] 1 dx = pi/2 => I = pi/4." }, { "id": "8fba3f5c-9ef5-4cdd-b90d-b93c74410530", "q": "What is the value of: Integral of x^2 * sin(x) dx from -pi to pi?", "options": ["0", "2 * pi^2", "pi/2", "1"], "answer": 0, "hint": "Product of an even function (x^2) and an odd function (sin x) is odd.", "explanation": "f(-x) = (-x)^2 * sin(-x) = -x^2 * sin(x) = -f(x). Since f(x) is odd and limits are symmetric, the integral is 0." }, { "id": "3eeda926-cf4b-490c-8696-be876beda308", "q": "Evaluate: Integral of |x|/x dx from -1 to 2.", "options": ["1", "2", "0", "3"], "answer": 0, "hint": "Split the integral into two regions: [-1, 0] where |x| = -x, and [0, 2] where |x| = x.", "explanation": "For [-1, 0], |x|/x = -x/x = -1. Integral from -1 to 0 of -1 dx is -1. For [0, 2], |x|/x = x/x = 1. Integral from 0 to 2 of 1 dx is 2. Total is -1 + 2 = 1." }], topicId: "2.2" })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = { ..._provideComponents(), ...props.components }; return MDXLayout ? _jsx(MDXLayout, { ...props, children: _jsx(_createMdxContent, { ...props }) }) : _createMdxContent(props); } return { default: MDXContent }; function _missingMdxReference(id, component) { throw new Error("Expected " + (component ? "component" : "object") + " `" + id + "` to be defined: you likely forgot to import, pass, or provide it."); }