Evaluating Definite Integrals Involving Inverse Trigonometric Functions

In this article, we will explore a method to evaluate a definite integral involving the inverse cotangent function. The integral in question is:

I ∫01 arccot(1 - x) x2 dx

Step-by-Step Evaluation

The notation we use for integrals is:

Figure 1: Integral Notation

Since the argument to the inverse cotangent function is positive, we can express the integrand in terms of the inverse tangent function. Applying the identity:

Figure 2: Inverse cotangent identity

We get:

Figure 3: Simplified integrand

The integrand can then be written as:

Figure 4: Simplified integration expression

Further simplification gives us:

Figure 5: Final integration expression

Using the interval-inverting substitution ( x 1 - x ), we find:

Figure 6: Substitution result

Hence, we can rewrite the integral ( I ) as:

Figure 7: Revised integral expression

Using integration by parts, where ( u 2 arctan{x} ) and ( dv dx ), we get:

Figure 8: Integration by parts expression

Evaluating the limits, we find:

Figure 9: Final result

Hence, the final value of the integral is:

Figure 10: Final answer boxed

Conclusion

The method to evaluate the given integral is a combination of expressing the integrand in terms of the inverse tangent function and applying integration by parts. This approach simplifies the integral and allows for a straightforward evaluation. The final value of the integral is ( frac{pi}{2} - ln{2} ).

Additional Insights

The integral evaluation process involves several key mathematical concepts:

Inverse trigonometric functions: Understanding the properties and identities of inverse cotangent and tangent functions. Interval-inverting substitutions: These substitutions help in simplifying the integral and making it easier to solve. Integration by parts: This technique is crucial for solving integrals that are not straightforward to evaluate.

By mastering these concepts, one can solve a wide range of definite integrals and enhance their problem-solving skills in calculus.

References

Fujiwara, M. (2012). Introduction to Analytic Number Theory. Springer. Ma, J. (2015). Calculus on Manifolds. Dover Publications.