Question

Solve each of the integrals in Exercises $75–78$by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.
$\int \frac{x^2-1}{x^2+1}dx$

Short answer

The value of integral is$x-2{\tan }^{-1}\left(x\right)+C$

Step-by-step solution

Step 1. Given informaion

An expression is given as$\int \frac{x^2-1}{x^2+1}dx$

Step 2. Solving integral

Split the term by using long distance method and simplify as,

$$\begin{gathered} \frac{x^2-1}{x^2+1}=1-\frac{2}{x^2+1} \\ \int \left(1-\frac{2}{x^2+1}\right)dx \\ =\int 1dx-2\int \frac{1}{x^2+1}dx \\ =x-2{\tan }^{-1}\left(x\right)+C \end{gathered}$$