Question

What is the antiderivative of `(2+x^2)/(1+x^2)`?

Answer 1

The Answer is `x+arctan(x)`

Explanation:

First note that : `(2+x^2)/(1+x^2)` can be written as `(1+1+x^2)/(1+x^2)=1/(1+x^2)+(1+x^2)/(1+x^2)=1+1/(1+x^2)`

`=>int(2+x^2)/(1+x^2)dx=int[1+1/(1+x^2)]dx=int[1]dx+int[1/(1+x^2)]dx=x+int[1/(1+x^2)]dx=`

The derivative of `arctan(x)` is `1/(1+x^2)`.

This implies that the antiderivative of `1/(1+x^2)` is `arctan(x)`

And it's on that basis that we can write : `int[1+1/(1+x^2)]dx=x+arctan(x)`

Hence,

`int(2+x^2)/(1+x^2)dx==int[1+1/(1+x^2)]dx=x+arctan(x) +c`

So the antiderivative of `(2+x^2)/(1+x^2)` is `color(blue)(x+arctan(x))`

`"NB :"`

Do not confuse the `antiderivative` with the indefinite integral

Antiderivative does not involve a constant. In fact finding the antiderivative doesn't mean intergrate!