Question

Integrate 3x^2/(x^2+1) dx.

Answer 1

Haha! Nice Hello Kitty pic! c:

Ok ok so here is one sneaky way you can approach this problem.\[\large \int\limits\frac{3x^2}{x^2+1}dx\]We'll start by adding and subtracting 3 from the top of the fraction. The purpose of this is to try and "create" an x^2+1 on top. If we can do so, we'll be able to simplify this fraction quite nicely.

\[\large \int\limits\limits\frac{3x^2+3-3}{x^2+1}dx\]Now let's factor a 3 out from the first 2 terms on top,\[\large \int\limits\limits\limits\frac{3(x^2+1)-3}{x^2+1}dx\]We did it! We formed an x^2+1 on top!
Now let's split this into two fractions.\[\large \int\limits\limits\limits\limits\frac{3(x^2+1)}{x^2+1}-\frac{3}{x^2+1}dx\]The first fraction will simplify very nicely!\[\large \int\limits 3-\frac{3}{x^2+1}dx\]Let's write this as two separate integrals just in case it's not clear enough already,\[\large 3\int\limits dx-3\int\limits \frac{1}{1+x^2}dx\]


I'll leave it to you to solve it from here! :)
That second integral is an important one to memorize.

Answer 2

Ah, thank you very much!