Substitution technique for integral

In summary, the conversation discusses integrating the function ∫ x5 √(x4 – 4) dx using substitution techniques and a table of integration. There is confusion about the hint given and whether to use integration by parts or trig substitution. Finally, a possible solution is suggested using integration by parts, but it is noted that the final result should not contain the variables u or a and the constant of integration should be included.
  • #1
mathaintmybag
4
0

Homework Statement



I need to use a substitution technique and then the table of integration to integrate the following:
∫ x5 √(x4 – 4) dx

and i am given a hint which is x5 = (x3)(x2)

I would assume that
u = x4 – 4, then du = 4x3
and that at some point a2 = 4 and a = 2

However, the x5 = (x3)(x2) is confusing the heck out of me. Need help!
 
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  • #2
What do a^2 and a have to do with anything?

The hint looks to me like it's guiding you to use integration by parts. Is that a technique that you have learned yet? I wouldn't advise it unless you have exhausted the possibilities for ordinary substitutions, however.
 
  • #3
What, so you have a limited table of integrals that doesn't include this and you are suggested a substitution to give you a form you can find in your table?

I think the point of the x5 = x3x2 is to use the x3 as you have suggested with your u = x4 - 4. You can substitute for the extra x2 like this:

x4 = u+4, x2 = sqrt(u+4)

This will give you a pure u integral that may be in your table.
 
  • #4
The reason i used a^2 = 4 and a = 2 is because in the book they used a number of different examples where they changed the number to a letter.

We have touched on integration by parts but everything we did learned involved using
ex so I am not really sure how to use it here.
 
  • #5
It can be done with trig substitution I believe. Setting x^4/4 to (x^2/2)^2 and substituting for cos(theta).

∫ x5 √(x4 – 4) dx

∫ x5 2√(x4/4 – 1) dx

∫ x5 2√((x2/2)^2 – 1) dx

d/dx cosΘ = d/dx x^2/2]
[2cosΘ] = x^2]
[4cos^2(Θ)] = x^4]

-sinΘ dΘ = 2x dx∫ x5 2√((x2/2)^2 – 1) dx

∫ x^4(2x)√((x2/2)^2 – 1) dx
∫4cos^2(Θ) (-sinΘ)sinΘ dΘAnd then from there it's pretty simple... Is that right?
 
Last edited:
  • #6
I haven't learned how to use trig substitutions yet so that will not help. Thanks for tip anyways.
 
  • #7
x5√(x4 - 4) = x2∙x3√(x4 - 4)
and you can use integration by parts on that.
 
  • #8
Would this be correct?
∫ x5 √(x4 – 4) dx
= ∫ ((x2*x2*x √(x4 – 4)) dx


Let u = x2, then du/2 = xdx
= ∫ 1/2 (u2√u2 – 4) du

Let a = 4 and a2 = 2
= ∫(u2√u2 – a2) du
= [x(u2 – a2)3/2 / 4] + [a2x√(u2 – a2) / 8] – (a4/8) ln(x + √(u2 – a2))
 
  • #9
It looks like you are using formula #10 here.

If so, you final result should not have u or a in it. You also need the constant of integration, which is needed in all indefinite integrals.
 

Related to Substitution technique for integral

1. What is substitution technique for integrals?

The substitution technique for integrals, also known as u-substitution, is a method used to solve integrals by substituting a variable in the integrand with another expression and then using the chain rule to simplify the integral.

2. When should I use substitution technique for integrals?

Substitution technique for integrals is useful when the integrand contains a function within another function, such as trigonometric or exponential functions. It can also be used to simplify integrals with complicated expressions.

3. How do I perform a substitution for integrals?

To perform a substitution for integrals, follow these steps:1. Identify a suitable substitution variable, usually denoted as u.2. Substitute u for the expression within the integral.3. Calculate the differential of u, du.4. Rewrite the integral in terms of u.5. Solve the integral using u.6. Substitute back the original expression in terms of u.

4. What is the purpose of substitution technique for integrals?

The purpose of substitution technique for integrals is to simplify complex integrals and make them easier to solve. It can also be used to transform integrals into a standard form that is easier to integrate.

5. Can I use any substitution variable for integrals?

Yes, any substitution variable can be used for integrals as long as it follows the rules of differentiation. However, it is recommended to use a variable that will simplify the integral and make it easier to solve.

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