How Do You Integrate ∫√((x^4)/2 + 9) dx?

  • Thread starter MusicDragon
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In summary: If you made a mistake on the original equation, your integrals will still be correct. Just change the variable and integrate.
  • #1
MusicDragon
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Homework Statement



∫√(((x)^2)/2 + 9) dx

Homework Equations



---

The Attempt at a Solution


I'm having a horrible time trying to do it. Problem is, I don't know how to start. The only thing I can think of is using trigonometric substitution. But I don't know how to manipulate the function in order to make it fit. A hint would be greatly appreciated.

*Edit* I made a mistake, it's
∫√(((x)^4)/2 + 9) dx I accidentally put it to the 2nd power instead to the 4th...
 
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  • #2
MusicDragon said:

Homework Statement



∫√(((x)^2)/2 + 9) dx


Homework Equations



---

The Attempt at a Solution


I'm having a horrible time trying to do it. Problem is, I don't know how to start. The only thing I can think of is using trigonometric substitution. But I don't know how to manipulate the function in order to make it fit. A hint would be greatly appreciated.
Trig substitution is the way to go.

Note that x2/2 = (x/√2)2.
Draw a right triangle with vertical leg labeled x/√2 and horizonatal leg labeled 3. Work out a relationship between the angle θ and x.

BTW, integration problems should not be posted in the Precalc section. I am moving this thread to the Calculus & Beyond section.
 
  • #3
Mark44 said:
Trig substitution is the way to go.

Note that x2/2 = (x/√2)2.
Draw a right triangle with vertical leg labeled x/√2 and horizonatal leg labeled 3. Work out a relationship between the angle θ and x.

I made a mistake on the equation to be integrated. It's to the power of 4, not 2. Does it still apply?
 
  • #4
x4/2 = (x2/√2)2

Same advice as before, except label one leg (the opposite would be my choice) as x2/√2 instead of x/√2.
 
  • #5
If it's sqrt(x^4/2+9) trig substitutions won't get you anywhere. You'd need special functions to do it.
 
  • #6
Dick said:
If it's sqrt(x^4/2+9) trig substitutions won't get you anywhere. You'd need special functions to do it.

Absolutely.
 

FAQ: How Do You Integrate ∫√((x^4)/2 + 9) dx?

1. What is the derivative of (((x)^2)/2 + 9)?

The derivative of (((x)^2)/2 + 9) is simply x.

2. How do you integrate (((x)^2)/2 + 9)?

To integrate (((x)^2)/2 + 9), you can use the power rule for integration, which states that the integral of x^n is equal to (x^(n+1))/(n+1). In this case, n = 2, so the integral is ((x^3)/3) + 9x + C.

3. What is the area under the curve of (((x)^2)/2 + 9) from x = 0 to x = 4?

The area under the curve of (((x)^2)/2 + 9) from x = 0 to x = 4 is equal to 36 square units.

4. Can you use substitution to solve the integral of (((x)^2)/2 + 9)?

Yes, you can use substitution to solve the integral of (((x)^2)/2 + 9). You can let u = x^2 and du = 2x dx, which will simplify the integral to (1/2)∫(u) du + 9∫dx. The first integral can be easily solved using the power rule, and the second integral is equal to 9x + C.

5. How can you verify the result of integrating (((x)^2)/2 + 9)?

You can verify the result of integrating (((x)^2)/2 + 9) by taking the derivative of your answer and seeing if it equals the original function. In this case, taking the derivative of ((x^3)/3) + 9x + C results in x, confirming that the integral was solved correctly.

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