Need help with integration by substitution for 9X^4 + 9X^2?

[superkam]

Homework Statement

Hi, I'm having trouble with the following problem:

$$\int \sqrt{9X^4 + 9X^2}$$

Homework Equations

Integration by substitution? U = some form of X

The Attempt at a Solution

Hi,
I assume that the best way to solve this integral is by using some sort of substitution, the problem is I don't exactly know what the substitute. I've tried the obvious option;$$U = 9X^4 + 9X^2$$ but didn't really get anywhere.
If anyone could give me any clues on what substitution to use I would really appreciate it.
Thanks in advance

Kam

 

[Dickfore]

$$\sqrt{9 x^{4} + 9 x^{2}} = \sqrt{9 x^{2} (x^{2} + 1)} = 3 x \sqrt{x^{2} + 1}$$

Then, you can either do the trigonometric substitution:

$$x = \tan{t}$$

or the hyperbolic substitution:

$$x = \sinh{t}$$

 

[Sourabh N]

x^2 + 1 = t also works well.

 

[superkam]

Okay, thank you very much for your help :)

 

[rwisz]

ila

Hi Kamila,

Thank you for reaching out for help with your integration problem. I am happy to assist you.

It looks like you have already attempted to use substitution by setting U = 9X^4 + 9X^2. However, this substitution may not be the most efficient way to solve this integral.

A helpful tip for choosing a substitution is to look for a function within the integral that has a derivative that is also present in the integral. In this case, both 9X^4 and 9X^2 have derivatives of 36X^3 and 18X, respectively.

So, let's try substituting U = 9X^2. Then, we have dU = 18X dX, which is present in the integral.

\int \sqrt{9X^4 + 9X^2} = \int \sqrt{9X^2(9X^2 + 1)} = \int \sqrt{U(U+1)}

Now, we can use the substitution U = 9X^2 to simplify the integral.

\int \sqrt{U(U+1)} = \int \sqrt{U} * \sqrt{U+1} dU

Using the power rule for integration, we can integrate \sqrt{U} to get \frac{2}{3}U^{\frac{3}{2}}.

So, our final solution is:

\int \sqrt{9X^4 + 9X^2} = \frac{2}{3} (9X^2)^{\frac{3}{2}} * \frac{2}{3}(9X^2 + 1)^{\frac{3}{2}} + C

= \frac{2}{3} (9X^2)^{\frac{3}{2}} * \frac{2}{3}(9X^2 + 1)^{\frac{3}{2}} + C

= \frac{2}{3} * 9^{\frac{3}{2}} * X^3 * \frac{2}{3} * (9X^2 + 1)^{\frac{3}{2}} + C

= 6X^3 * \frac{2}{3} * (9X^2 + 1)^{\frac{3}{2}} + C

=