Primitive of x/sqrt(4+x^4): Is Substitution Necessary?

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In summary, the conversation discusses solving a primitive using the substitution method and the use of inverse hyperbolic integrals. The conversation also explores the possibility of solving the primitive without making any substitutions, by manipulating the given function.
  • #1
esmeco
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I know that tu solve this primitive we have tu use the substitution method,but I think that none of the rules that should be used apply to this!The problem is, to use the substitution: x=a/b sen t we should have the function in the format (sqrt(a^2 - b^2*x^2)),but instead of a x^2 I have a x^4.
I'm studying for an exam about primitives I'm having tomorrow and I really could use some help on this!
Thanks in advance for the reply!
 
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  • #2
Let u = x^2
du/dx = 2x
du = 2xdx

Hence 1/2 Integral (1/ Root( u^2 + 4) ) du
Which is a inverse hyperbolic integral.
 
  • #3
Make the substitution [itex] x^2 =t [/itex].

Daniel.
 
  • #4
Now that i look at the primitive,I think it's not necessary to make substitutions...Multyplying the fraction by 1/4 it could be something like: 1/4*x/sqrt(1/4 + (1/4x)^4),which in turn would loook like: 1/4primitive x/sqrt(1/4 + (1/2x^2)2).Since it's no the form f'/sqrt(1-f^2) , the primitive of the function would be: 1/4arcsen(1/2x^2).
Could this be solved this way?
Thanks in advance for the reply!^_^
 

Related to Primitive of x/sqrt(4+x^4): Is Substitution Necessary?

1. What is the formula for the primitive of x/sqrt(4+x^4)?

The formula for the primitive of x/sqrt(4+x^4) is √(4+x^4) + C.

2. How do you solve for the primitive of x/sqrt(4+x^4)?

To solve for the primitive of x/sqrt(4+x^4), you can use the substitution method by letting u = x^2 and du = 2x dx. This will result in the integral becoming 1/2 ∫ du/√(4+u^2), which can then be solved using trigonometric substitution.

3. Can the primitive of x/sqrt(4+x^4) be simplified?

Yes, the primitive of x/sqrt(4+x^4) can be simplified to √(4+x^4) + C by using the substitution method or integration by parts.

4. What is the domain of the primitive of x/sqrt(4+x^4)?

The domain of the primitive of x/sqrt(4+x^4) is all real numbers except for x = ±√2. This is because the integrand becomes undefined when x = ±√2.

5. How does the primitive of x/sqrt(4+x^4) relate to the original function?

The primitive of x/sqrt(4+x^4) is the anti-derivative of the original function, which means that when the primitive is differentiated, it will result in the original function. This relationship is known as the fundamental theorem of calculus.

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