The integral of (4x+3)/(x^2+1) - Version 2

In this video, I demonstrate how to integrate (or find the antiderivative of) the expression (4x+3)/(x²+1) by using a trigonometric substitution.

Whenever we have a term that has the form of x²+a² in the denominator, in this case, we have x²+1 = x²+1², the appropriate substitution we should make is:

x = atanθ

So, since in this case a = 1, the substitution we should make is x=tanθ.

Now, the derivative of this substitution with respect to θ is: dx=sec²θdθ

We all that, the integral becomes:

\int \frac{4x+1}{x^2+1}\mathrm{d}x=\int \frac{4\tan \theta+3}{\tan^2+1}\sec^2\theta\mathrm{d}\theta=\int 4\tan \theta+3\mathrm{d}\theta