Integrating $\int^u_0\frac{dx}{x^2+3x+2}$

Question

Find $$\int^u_0\frac{dx}{x^2+3x+2}$$

I'm having trouble making progress on this. Specifically, I know the following means to find an integral:

1. Antiderivative
2. Integration by parts
3. Change of variable
4. Definition of integral (either Darboux or Riemann).

None of these have been helpful to me:

1. $\frac{1}{x^2 + 3x + 2}$ is not the derivative of anything I can construct. The closest is $\frac{2x + 3}{x^2 + 3x + 2} = [\log (x^2 + 3x + 2)]'$. This might be useful in combination with another method, so let's keep on looking.
2. I've tried various combinations, such as $u = \frac{1}{x^2 + 3x + 2}, v= x^2 + 3x$ or $u = \log(x^2 + 3x + 2), v = \frac{1}{2x+3}$. Again, none have gotten me further.
3. $f(x) = 1/x, u(x) = x^2 + 3x +2, u'(x) = 2x$. The problem is I can't get the $u'(x)$ factor to appear.
4. I've made no progress with the definitions. In fact, I've only seen definitions used to prove general theorems, never to compute a specific integral.

Can you point me in the right direction? Please do not post the full solution; just let me know how I need to tackle this.

Answer 1

HINT

You can factor the denominator in order to obtain:

\begin{align*} \frac{1}{x^{2} + 3x + 2} & = \frac{1}{(x + 1)(x + 2)}\\\\ & = \frac{(x+2) - (x+1)}{(x+1)(x+2)}\\\\ & = \frac{1}{x+1} - \frac{1}{x+2} \end{align*}

Can you take it from here?

Answer 2

I will just provide the Ansatz for partial fraction decomposition.

You factor by looking for the roots of $x^2+3x+2$, which might be $x_1$ and $x_2$. I use completion of the square.

Then, if we got two different roots $x_i$, you set $$ \frac{1}{x^2+3x+2} = \frac{1}{(x-x_1)(x-x_2)} = \frac{C}{x-x_1} + \frac{D}{x-x_2} $$ and multiply both sides by $(x-x_1)(x-x_2)$ and sort by powers of $x^0 = 1$ and $x^1 = x$ to get a linear system of two equations in the unknowns $C$ and $D$.

After you determined $C$ and $D$ you can continue to integrate the easier two integrals.