Integrate (x^3)/sqrt(4x^(2)+9)^(3): Steps & Tips

In summary, the conversation discussed how to integrate (x^3)/sqrt(4x^(2)+9)^(3) dx using different methods such as u-substitution, trig-substitution, and integration by parts. It was suggested to break up the x^3 term and make a substitution, or use a trigonometric substitution. A link was also provided to Wolfram|Alpha, which showed the steps for the integration.
  • #1
SJB3415
1
0
How do you integrate: (x^3)/sqrt(4x^(2)+9)^(3) dx

I have done integration with sqrts before but not when it is cubed up the square root...Any suggestions?
 
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  • #2
How can you remove the [tex]x^3[/tex] term?
 
  • #4
just to make sure, is this your problem?
[tex]\int \frac{x^3}{(4x^2+9)^{3/2}} \,dx[/tex]

i see two possible approaches to this problem. although this doesn't look like a typical u-substitution problem, try making one. hint: you usually let the u be the thing inside a composition of functions, especially if it's underneath a square root. so, try letting u=4x2+9. then you have the problem of accounting for the x3. break x3 up into x2x. then you'll need to find x2 in terms of u. after all this, it should be a simple integral to finish.

the other method (that i haven't worked out) would be to try a trig-substition. in this case it would be [itex]x=\frac{3}{2}\tan\theta[/itex]. like i said, i didn't take the time to work this out, but this should work if you can finish the trig integral you'll be left with after making the substitution.

FanofAFan said:
You could try pluging in the problem in Wolfram|alpha...

I actually did for you... follow the link, they work it out in complete steps
http://www.wolframalpha.com/input/?i=(x^3)/sqrt(4x^(2)+9)^(3)

no where in that link did they actually work this problem out in complete steps. wolframalpha is only helpful for checking your answers.
 
  • #5
where it says on the right side in orange letter, "Show sets" under integration
 
  • #6
FanofAFan said:
where it says on the right side in orange letter, "Show sets" under integration

my apologies, and i stand corrected. although the steps could be seen as confusing. they actually make the substitution that i mentioned above, but they do so using two separate substitutions.
 

FAQ: Integrate (x^3)/sqrt(4x^(2)+9)^(3): Steps & Tips

What is the meaning of integration?

Integration is a mathematical process that involves finding the area under a curve. It is the inverse operation of differentiation, which involves finding the slope of a curve at a given point.

What is the general process for integrating a function?

The general process for integrating a function is to first identify the type of function and then use appropriate integration techniques, such as substitution, integration by parts, or trigonometric substitution, to find the antiderivative of the function. The antiderivative is the function that, when differentiated, yields the original function.

What is the function we are trying to integrate and why is it challenging?

The function we are trying to integrate is (x^3)/sqrt(4x^(2)+9)^(3). This function is challenging because it contains a radical and a polynomial term, which require different integration techniques. Additionally, the presence of the variable x in both the numerator and denominator makes it difficult to find a suitable substitution.

What is the step-by-step process for integrating (x^3)/sqrt(4x^(2)+9)^(3)?

One possible step-by-step process for integrating (x^3)/sqrt(4x^(2)+9)^(3) is as follows:

  1. Use the substitution method by letting u = 4x^2 + 9 and du = 8x dx.
  2. Rewrite the original function in terms of u, giving us (1/8)∫(u-9)/sqrt(u)^3 du.
  3. Use the power rule for integration to simplify the function to (1/8)∫u^(-5/2) - 9u^(-3/2) du.
  4. Use the sum rule for integration to split the integral into two separate integrals.
  5. Use the power rule and the constant multiple rule to integrate each term, giving us (1/8)(-2/3)u^(-3/2) - (1/8)(2/5)u^(-5/2) + C.
  6. Substitute back in the original variable x and simplify the expression.

What are some tips for successfully integrating (x^3)/sqrt(4x^(2)+9)^(3)?

Here are some tips for successfully integrating (x^3)/sqrt(4x^(2)+9)^(3):

  • Try different substitution techniques until you find one that simplifies the function.
  • Remember to apply any necessary rules for integration, such as the power rule or the sum rule.
  • Be careful with signs and constants when substituting back in the original variable.
  • Practice and review different integration techniques to improve your skills.

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