Evaluating Integrals Involving the Inverse Tangent Function: Techniques and Recursive Formulas

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The inverse tangent function, often denoted as ( text{arctan}(x) ) or ( tan^{-1}(x) ), appears frequently in calculus and mathematical analysis. This article explores various techniques for evaluating integrals that involve this function, including integration by parts, recursive formulas, and detailed step-by-step solutions. We will also delve into the derivation of recursive relationships that enable the simplification of such integrals.

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Integration by Parts

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One of the most common methods for integrating a product of functions is integration by parts. This technique states that for functions ( u(x) ) and ( v(x) ), the integral ( int u , dv ) can be expressed as ( uv - int v , du ).

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Example: ( int x^3 arctan(x) , dx )

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Let's evaluate the integral ( int x^3 arctan(x) , dx ) using integration by parts.

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Choose ( u arctan(x) ) and ( dv x^3 , dx ).

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Then, ( du frac{1}{x^2 1} , dx ) and ( v frac{x^4}{4} ).

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Applying integration by parts:

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[ int x^3 arctan(x) , dx frac{x^4}{4} arctan(x) - int frac{x^4}{4} cdot frac{1}{x^2 1} , dx ]

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Simplify the integral:

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[ frac{x^4}{4} arctan(x) - frac{1}{4} int frac{x^4 - 1}{x^2 1} , dx ]

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Continue the integration:

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[ frac{x^4}{4} arctan(x) - frac{1}{4} int left( x^2 - 1 right) cdot frac{1}{x^2 1} , dx ]

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[ frac{x^4}{4} arctan(x) - frac{1}{4} left( frac{x^3}{3} - x arctan(x) right) C ]

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[ frac{1}{4} left( x^4 - 1 right) arctan(x) - frac{1}{12} x^3 C ]

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Thus, the integral ( int x^3 arctan(x) , dx ) equals:

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[ boxed{frac{1}{4} left( x^4 - 1 right) arctan(x) - frac{1}{12} x^3 C} ]

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Recursive Integration Formulas

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To handle more complex integrals involving ( arctan(x) ), we can derive recursive formulas. Consider the general integral:

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[ I_n int x^{n-1} arctan(x) , dx ]

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Using integration by parts, we can express this integral recursively. Let:

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[ u arctan(x), quad dv x^{n-1} , dx ]

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[ du frac{1}{x^2 1} , dx, quad v frac{x^n}{n} ]

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[ I_n frac{x^n}{n} arctan(x) - int frac{x^n}{n} cdot frac{1}{x^2 1} , dx ]

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[ frac{x^n}{n} arctan(x) - frac{1}{n} int frac{x^2 - 1}{x^2 1} , dx ]

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[ frac{x^n}{n} arctan(x) - frac{1}{n} left( int frac{x^2}{x^2 1} , dx - int frac{1}{x^2 1} , dx right) ]

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[ frac{x^n}{n} arctan(x) - frac{1}{n} left( x - arctan(x) right) - frac{1}{n} arctan(x) ]

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[ frac{x^n}{n} arctan(x) - frac{x}{n} C ]

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Recursive Formula for ( I_n )

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For a general ( I_n ), the recursive formula is given by:

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[ I_n frac{x^n}{n} arctan(x) - frac{1}{n} left( x^{n-2} arctan(x) - int x^{n-2} cdot frac{1}{x^2 1} , dx right) C ]

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Using the duality of even and odd exponent integrals, we can derive a simplified recursive formula:

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[ I_n frac{x^n}{n} arctan(x) - frac{1}{n} left( x^{n-2} arctan(x) - I_{n-2} right) C ]

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[ I_n frac{x^n}{n} arctan(x) - frac{x^{n-2} arctan(x)}{n} frac{1}{n} I_{n-2} C ]

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[ I_n frac{1}{n} left( x^n - x^{n-2} right) arctan(x) frac{1}{n} I_{n-2} C ]

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Conclusion

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Integration techniques such as integration by parts and recursive formulas provide powerful tools for evaluating integrals involving the inverse tangent function. These methods not only simplify complex integrals but also reveal intricate patterns and relationships in mathematical analysis. Applying these techniques allows for a deeper understanding of the behavior of the inverse tangent function and its interactions with other functions.