How To Integrate 1/[sqrt (x^2 + 3x + 2)] dx?

[vela]

Take a break and try again later with fresh eyes.

 

[PeroK]

There's a less tricky way to integrate that.
$$\int \sec \theta\,d\theta = \int \frac 1{\cos \theta}\,d\theta = \int \frac {\cos\theta}{\cos^2 \theta}\,d\theta = \int \frac {\cos\theta}{1-\sin^2 \theta}\,d\theta$$ then use $u=\sin\theta$.

Using $\cosh$ is about a hundred million time better. If you'll excuse the hyperbole!

 

[askor]

Using $\cosh$ is about a hundred million time better. If you'll excuse the hyperbole!

How do you use $\cosh$? Please show me an example.

 

[Mark44]

How do you use $\cosh$? Please show me an example.

Have the hyperbolic trig functions been presented to you, yet? If not, an example probably won't make much sense.

 

[Vanadium 50]

If you'll excuse the hyperbole!

I saw what you did there.

 

[Charles Link]

and even what @PeroK presented in post 12 takes a little algebra to compute, starting with $x=\cosh{u}=\frac{e^u+e^{-u}}{2}$.

 

[askor]

I only know the very basic properties of hyperbolic function such as:

$\sinh x = \frac{e^x - e^{-x}}{2}$

and

$\cosh x = \frac{e^x + e^{-x}}{2}$

But I don't know how to use it in technique of integration.

 

[PeroK]

I only know the very basic properties of hyperbolic function such as:

$\sinh x = \frac{e^x - e^{-x}}{2}$

and

$\cosh x = \frac{e^x + e^{-x}}{2}$

But I don't know how to use it in technique of integration.

Using the above definitions what are the derivatives of $\sinh x$ and $\cosh x$?

What is $\cosh^2 x$ in terms of $\sinh^2 x$?

In terms of integration, you use them the same way you use the trig functions, by substitution. E.g.:
$$x = \cosh u, \ \ dx = \frac{d}{du} (\cosh u) du$$

 

[trees and plants]

I only know the very basic properties of hyperbolic function such as:

$\sinh x = \frac{e^x - e^{-x}}{2}$

and

$\cosh x = \frac{e^x + e^{-x}}{2}$

But I don't know how to use it in technique of integration.

I think finding $dx$ helps so you need to find the derivative of $\cosh x$. I learned it from a table of derivatives and integrals, which contained also $\cosh x$.

 

[octopus26]

How do you integrate $\frac{1}{\sqrt{x^2 + 3x + 2}} dx$?

I had tried using $u = x^2 + 3x + 2$ and trigonometry substitution but failed.

Please give me some clues and hints.

Thank you

mentor note: moved from a non-homework to here hence no template.

Wolfram Alpha

 

[haruspex]

$\int \frac{du}{1 - u^2}$

The hyperbolic trig approach is neater, but to proceed with the above form use partial fractions.

 

[Grasshopper]

The hyperbolic trig approach is neater, but to proceed with the above form use partial fractions.

With you have there, why couldn’t a second substitution be made, using $1 - sin^2w = cos^2w$?

I’m sure I’m missing something.

 

[haruspex]

With you have there, why couldn’t a second substitution be made, using $1 - sin^2w = cos^2w$?

I’m sure I’m missing something.

That would be going back to what we had earlier, integrating sec.
Do you see how to solve it using partial fractions?

 

[Grasshopper]

That would be going back to what we had earlier, integrating sec.
Do you see how to solve it using partial fractions?

Wait, lol sorry I see now. It was late I was not thinking clearly. But yeah this is one of the easier partial fractions.

 

[chwala]

Wait, lol sorry I see now. It was late I was not thinking clearly. But yeah this is one of the easier partial fractions.

what is the final solution? did you really get it?