I need to integrate, but I don't know where to start. I'm a bit (a lot) rusty.

Question

Please help me solve the following integral.

I'm not sure where to start in fact, I haven't done math for 7 years and now I have to jump back in the integrals. I'm not sure if I have to integrate by part or by substitution. Wolfram Alpha tells me to start with the long division, but I'm not quite sure what it means. I would like to give you guys more information, but the thing is, I don't even know where to start, so it's kind of hard to give more details. Could you please at least tell me where to start? I'm looking in my manuals, and I just can't find it. Thank you very much!

$$ \int\frac{3x^3+2x+7}{x^2+3}dx $$

Answer 1

As you already mentioned you can start by "long division":

$$\int \frac{3x^3+2x+7}{x^2+3}dx=\int 3x-\frac{7(x-1)}{(x^2+3)} \space \space dx$$

For a good video on polynomial division see here: LINK

Now you can separate your integral:

$$-7\int \frac{(x-1)}{(x^2+3)} \space dx+3\int x \space dx=\color{red}{-7\int \frac{x}{x^2+3} \space dx}+\color{blue}{7\int \frac{1}{x^2+3}\space dx}+3\int x \space dx $$

I am assuming you know how to integrate the last integral. The red and blue integral can be solved by substitution.

Answer 2

Hint:

Combine these fractions:

$3x-\frac{7}{2}\frac{2x}{x^2+3}+\frac{7}{x^2+3}$

Polynomial Division is simply the reversal of this process