त्रिकोणमितीय प्रतिस्थापन

गणित के सन्दर्भ में, त्रिकोणमितीय प्रतिस्थापन (Trigonometric substitution) का अर्थ है, गैर-त्रिकोणमितीय फलनों के स्थान पर त्रिकोणमितीय फलनों को स्थापित करना। इनके उपयोग से कुछ समाकल सरल हो जाते हैं।Stewart James James Stewart (mathematician) Calculus: Early Transcendentals Brooks/Cole 6th 2008 0-495-01166-5 registration https://archive.org/details/calculusearlytra00stew_1 Thomas George B. Weir Maurice D. Hass Joel Joel Hass George B. Thomas Thomas' Calculus: Early Transcendentals https://archive.org/details/thomascalculusea0000weir_s0z8 Addison-Wesley 2010 12th 0-321-58876-2 प्रतिस्थापन 1. यदि समाकल्य (integrand) में a2 − x2 हो तो , रखें और यह सर्वसमिका प्रयोग करें- प्रतिस्थापन 2. If the integrand contains a2 + x2, let and use the identity प्रतिस्थापन 3. If the integrand contains x2 − a2, let and use the identity उदाहरण Integrals containing a2 − x2 In the integral we may use Note that the above step requires that a > 0 and cos(θ) > 0; we can choose the a to be the positive square root of a2; and we impose the restriction on θ to be −π/2 < θ < π/2 by using the arcsin function. For a definite integral, one must figure out how the bounds of integration change. For example, as x goes from 0 to a/2, then sin(θ) goes from 0 to 1/2, so θ goes from 0 to π/6. Then we have Some care is needed when picking the bounds. The integration above requires that −π/2 < θ < π/2, so θ going from 0 to π/6 is the only choice. If we had missed this restriction, we might have picked θ to go from π to 5π/6, which would give us the negative of the result. Integrals containing a2 + x2 In the integral we may write so that the integral becomes (provided a ≠ 0). सन्दर्भ इन्हें भी देखें समाकलन चर परिवर्तन प्रतिस्थापन द्वारा समाकलन श्रेणी:समाकलन