Integral of 1 Over Square of Sum of Squares

Evaluate $\displaystyle \int \frac{1}{(x^2 + 1)^2} \, dx$ using an appropriate substitution method.

This problem requires the evaluation of an integral involving the square of a sum of squares in the denominator, specifically $\int \frac{1}{(x^2 + 1)^2} \, dx$. To solve this integral, a good starting point is recognizing the structure of the expression. It might remind one of forms that are typically encountered when employing trigonometric substitution due to the $x^2 + 1$ term which is suggestive of the tangent trigonometric identity. However, the squared nature of the denominator indicates that a straightforward trigonometric substitution might become algebraically cumbersome.

The alternative is to consider a rational trigonometric substitution or a more sophisticated algebraic technique that simplifies the integral without introducing excessive algebraic complexity. One possible method is using partial fraction decomposition if applicable, or recognizing it as a special form in standard integral tables. It's crucial to identify patterns or substitutions that transform the integral into a more recognizable or simpler form.

Students addressing this problem should have a good grasp of recognizing integral forms that suggest substitution methods, especially those relating to $x^2 + a^2$ forms. They should also be comfortable manipulations involving algebraic identities and be aware of strategies to simplify quadratic expressions. The problem is moderately challenging, as it requires an ability to discern the most efficient method of attacking the given integral form.

Related Problems