How can I solve this integral using integration by parts?

In summary: Thanks, yea now i can solve it, i got it.In summary, the conversation is about a problem involving the integration of x^2 sin(x^2) dx. The participants discuss using the substitution method and integration by parts, but ultimately conclude that the integrand does not have an elementary antiderivative. It is later revealed that the problem was a typo and should have been x^2 sin^2(x) instead. The conversation also includes a discussion on how to type the integral sign in LaTeX.
  • #1
jacy
76
0
Hi ,
I am having trouble integrating this problem.

Integrate x^2 sin(x^2) dx

Here is what i did. I used the substitution method.

u = x^2 sqrt u = x
du = 2x dx
du/2x = dx

du/2 sqrt u = dx since sqrt u = x

substituting this in the equation

u sin(u) du/2 sqrt u

Now i don't know how to integrate this. Please help, thanks.
 
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  • #2
jacy said:
Hi ,
I am having trouble integrating this problem.

Integrate x^2 sin(x^2) dx

I don't think the antiderivative of your integrand is an elementary...but I don't really know for sure.
 
  • #3
I believe you're correct. My computer cannot calculate it.
 
  • #4
apmcavoy said:
I believe you're correct. My computer cannot calculate it.


Thanks 4 ur time. Do u think that problem is wrong.
 
  • #5
jacy said:
Thanks 4 ur time. Do u think that problem is wrong.

The problem is fine...it's just that you cannot find an elementary antiderivative as a solution.
 
  • #6
where'd you obtain the problem?
 
  • #7
It would work if the integrand was [itex]x\sin{x^{2}}[/itex].
 
  • #8
GCT said:
where'd you obtain the problem?

Thanks for looking at the problem. My teacher gave this problem on the exam. How can we solve this.
 
  • #9
Integration by parts might be possible, but my computer and my TI 89 cannot compute it, so there's probably not a simple antiderivative. And you were given this problem on a test? :bugeye:
 
  • #10
Tony11235 said:
Integration by parts might be possible, but my computer and my TI 89 cannot compute it, so there's probably not a simple antiderivative. And you were given this problem on a test? :bugeye:

Thanks, do u think the substitution method that i used will work. This wasn't the only one there were 3 more.
 
Last edited:
  • #11
Are you sure it wasn't asking you to evaluate it as a definite integral numerically? That's the only way I can see doing this.
 
  • #12
apmcavoy said:
Are you sure it wasn't asking you to evaluate it as a definite integral numerically? That's the only way I can see doing this.

No it isn't a definite integral. It's tough one. Hopefully the teacher should provide the solution today. If he does, then can i post the solution in here, thanks.
 
  • #13
[itex] \int x^2\sin(x^2) dx [/tex]

[itex] \int x^2\sin(x^2) dx = \frac{-x}{2}\cos(x^2) + \frac{1}{2} \int \cos(x^2) dx [/tex]
 
  • #14
whozum said:
[itex] \int x^2\sin(x^2) dx [/tex]

[itex] \int x^2\sin(x^2) dx = \frac{-x}{2}\cos(x^2) + \frac{1}{2} \int \cos(x^2) dx [/tex]

True, but now what's [itex]\frac{1}{2} \int \cos(x^2) dx[/itex]?

jacy, the initial response you received was correct. Your integrand doesn't have an elementary antiderivative. 10 bucks says that the problem was supposed to be [itex]\int x\sin(x^2) dx[/itex].
 
  • #15
Tom Mattson said:
True, but now what's [itex]\frac{1}{2} \int \cos(x^2) dx[/itex]?

jacy, the initial response you received was correct. Your integrand doesn't have an elementary antiderivative. 10 bucks says that the problem was supposed to be [itex]\int x\sin(x^2) dx[/itex].

Its not elementary i was just trying it out and seeing how far I could get, but figured I'd just post it anyway.
 
  • #16
Here's the answer from Mathematica in case anyone was wondering. I agree, it was probably supposed to be [tex]\int x\sin{(x^2)}dx[/tex].
 

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  • #17
Tom Mattson said:
True, but now what's [itex]\frac{1}{2} \int \cos(x^2) dx[/itex]?

jacy, the initial response you received was correct. Your integrand doesn't have an elementary antiderivative. 10 bucks says that the problem was supposed to be [itex]\int x\sin(x^2) dx[/itex].

I don't think so it was a typo. This problem was on the exam, thanks.
 
  • #18
jacy said:
I don't think so it was a typo. This problem was on the exam, thanks.

Was the problem asking to find the derivative of that function?
 
  • #19
whozum said:
Was the problem asking to find the derivative of that function?

We have to find the anti derivative of that function. Today the teacher said that he made a typo it should be
integrate x^2 sin^2(x) instead of x^2 sin(x^2)

How did u guys type the sign of integral
 
  • #20
[tex] \int {x^2 \sin ^2 x\,dx} [/tex]
can be evaluated via integration by parts.

jacy, the integral sign is just \int when using LaTeX.
Try clicking on this integral sign: [tex] \int [/tex]
 
  • #21
bomba923 said:
[tex] \int {x^2 \sin ^2 x\,dx} [/tex]
can be evaluated via integration by parts.

jacy, the integral sign is just \int when using LaTeX.
Try clicking on this integral sign: [tex] \int [/tex]

Thanks, yea now i can solve it
 

1. What is the general formula for integrating x^2 sin(x^2) dx?

The general formula for integrating x^2 sin(x^2) dx is ∫x^2 sin(x^2) dx = -1/2 cos(x^2) + C.

2. How do you integrate x^2 sin(x^2) dx using substitution?

To integrate x^2 sin(x^2) dx using substitution, let u = x^2. Then du = 2x dx and the integral becomes ∫sin(u) du. This can then be integrated as -cos(u) + C, and substituting back in for u gives the final answer of -cos(x^2) + C.

3. What is the method for integrating x^2 sin(x^2) dx using integration by parts?

The method for integrating x^2 sin(x^2) dx using integration by parts involves choosing u = x^2 and dv = sin(x^2) dx, and then applying the formula ∫u dv = uv - ∫v du. This results in the integral becoming -1/2 cos(x^2) + 1/2∫cos(x^2) dx, which can then be integrated using substitution or by recognizing it as the sine integral function.

4. Can the integral of x^2 sin(x^2) dx be solved using trigonometric identities?

Yes, the integral of x^2 sin(x^2) dx can also be solved using trigonometric identities such as the double angle formula for sine. This results in the integral becoming ∫(1/2)(1-cos(2x^2)) dx, which can then be integrated using substitution or by recognizing it as the sine integral function.

5. What are some real-world applications of integrating x^2 sin(x^2) dx?

The integral of x^2 sin(x^2) dx has various applications in physics, engineering, and mathematics. For example, it can be used in calculating the electric field of a charged particle, the motion of a pendulum, or the intensity of light after passing through a diffraction grating. It also has applications in signal processing and Fourier analysis.

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