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।

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

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