# Integral Solving?

$$\int$$$$\frac{x+5}{2x^2+2x+3}$$

I would be very greatfull. :D

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
The first thing to do is to try to factor the denominator, then use "partial fractions".
Using the quadratic formula, the roots of $2x^2+ 2x+ 3= 0$ are
$$\frac{-2\pm\sqrt{2^2- 4(2)(3)}}{4}= \frac{-2\pm\sqrt{-24}}{5}[/itex] This has no real roots so certainly cannot be factored with real coefficients. So complete the square in the denominator: [tex]2x^2+ 2x+ 3= 2(x^2+ x)+ 3$$
$$2(x^2+ x+ 1/4- 1/4)+ 3= 2(x^2+ x+ 1/4)- 1/2+ 3[/itex] [tex]= 2(x+ 1/2)^2+ 5/2[/itex] Try the substitution u= x+ 1/2 and remember that [tex]\int\frac{dx}{x^2+ 1}= arctan(x)+ C$$

Cyosis
Homework Helper
Would be nice if you actually bothered to attempt it and show us your attempts. You could try completing the square.

Thanks for the answears.
I havent got any time at all , tomorow i have the exam and i am studying some other exercises.
I havent done much efforts on this one but i am asking you if you have got some time to solve it.I know it sounds strange but i still hope somebody with a god will would help.
Another thing i wanted to ask : is there any integral solver online.(i want that to confirm the integrals i have solved). Thank you again.

ideasrule
Homework Helper
$$\int\frac{(u+9/2)du}{2u^2+5/2}= \int\frac{u du}{2u^2+5/2}+\int\frac{9/2 du}{2u^2+5/2}$$

The first integral should be easy; just make the substitution w=2u^2+5/2. For the second integral:

$$\int\frac{9/2 du}{2u^2+5/2}=\int\frac{9/5du}{4/5u^2+1}$$

Make the substitution w=sqrt(4/5u^2)=u*sqrt(4/5), use $$\int\frac{dx}{x^2+1}=arctan(x)$$, and everything else should be easy.

As for an online integrator, try http://integrals.wolfram.com/index.jsp?expr=(x%2B5)%2F(2x^2%2B2x%2B3)&random=false

Cyosis
Homework Helper
Penultimate said:
Thanks for the answears.
I havent got any time at all , tomorow i have the exam and i am studying some other exercises.
I havent done much efforts on this one but i am asking you if you have got some time to solve it.I know it sounds strange but i still hope somebody with a god will would help.
Another thing i wanted to ask : is there any integral solver online.(i want that to confirm the integrals i have solved). Thank you again.

You check your answers by differentiating them.