# Thread: What kind of integral, and how do i start?

1. ## What kind of integral, and how do i start?

∫5/x(x²+2x+5)dx

Is this integration by parts?

2. Originally Posted by hlpplz
∫5/x(x²+2x+5)dx

Is this integration by parts?
This will be a partial fractions problem.

Note that $\displaystyle x^2+2x+5$ is an irreducible quadratic.

Thus, $\displaystyle \frac{5}{x(x^2+2x+5)}=\frac{A}{x}+\frac{Bx+C}{x^2+ 2x+5}$

Thus, we see that $\displaystyle 5=A(x^2+2x+5)+Bx^2+Cx\implies 5=(A+B)x^2+(2A+C)x+5A$

Comparing coefficients, we have $\displaystyle A+B=0$, $\displaystyle 2A+C=0$ and $\displaystyle 5A=5\implies A=1$

Now, we see that $\displaystyle C=-2A=-2(1)=-2$ and $\displaystyle B=-A=-1$

So $\displaystyle \frac{5}{x(x^2+2x+5)}=\frac{1}{x}-\frac{x+2}{x^2+2x+5}$

Therefore, $\displaystyle \frac{5\,dx}{x(x^2+2x+5)}=\int\frac{\,dx}{x}-\int\frac{x+2}{x^2+2x+5}\,dx$

Can you take it from here?

3. i don't know what to do with the second integral.
I know the first one is simply ln⎮x⎮. Do i have to integrate by partial fractions the second integral again?

4. Originally Posted by hlpplz
i don't know what to do with the second integral.
I know the first one is simply ln⎮x⎮. Do i have to integrate by partial fractions the second integral again?

Yes.

ln⎮x⎮- 1/2ln⎮x²+2x+5⎮- 1/x+1tan-¹2/x+1 + C ?

6. Originally Posted by hlpplz
∫5/(x(x²+4x+5))dx

I got an answer of: ln⎮x⎮- 1/2ln⎮x²+4x+5⎮- 1/(x+1) tan-¹2/(x+1) + C
Is this correct?

7. i havent done integrals in a long time but i just took a shot at this and i think you made a mistake

8. how would i solve it?

9. Originally Posted by hlpplz
∫5/(x(x²+4x+5))dx

I got an answer of: ln⎮x⎮- 1/2ln⎮x²+4x+5⎮- 1/(x+1) tan-¹2/(x+1) + C
Is this correct?
It doesn't look right. If you post all your working it will be easier to find your mistake(s).

10. Here is the work i did

11. error in this step
$\displaystyle \int {\frac{da}{a^2 +4}}$

= $\displaystyle \frac{1}{2}tan^{-1}(\frac{2}{a}) ~+c$

12. other than that all is well?

13. Originally Posted by hlpplz
other than that all is well?
Yes My dear , you are absolutely corrrrecttt... Cool Job buddy