How do you test the improper integral #int x^3 dx# from #(-oo, oo)# and evaluate if possible?

Answer 1

The integral is not convergent as:

#int_(-oo)^(+oo) x^3 dx = lim_(u->oo) int_(-u)^0 x^3dx + lim_(v->oo) int_0^v x^3dx#
#int_(-oo)^(+oo) x^3 dx = lim_(u->oo) -u^4/4 + lim_(v->oo) v^4/4#

The two limits should be finite separately and they are not.

The integral is however convergent in the sense of Cauchy's principal values as #x^3# is an odd function, so:
#int_(-t)^t x^3dx = [x^4/4]_(-t)^t = 0#

and then:

#lim_(t->oo) int_(-t)^t x^3dx = 0#
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Answer 2

To test the improper integral ( \int_{-\infty}^{\infty} x^3 , dx ) for convergence or divergence, you can split it into two separate integrals, one from ( -\infty ) to ( 0 ) and the other from ( 0 ) to ( \infty ). Then, evaluate each integral individually.

  1. For the integral from ( -\infty ) to ( 0 ):

[ \int_{-\infty}^{0} x^3 , dx ]

Apply the antiderivative:

[ \left[ \frac{x^4}{4} \right]_{-\infty}^{0} ]

Since ( x^4 ) grows without bound as ( x ) approaches negative infinity, this integral diverges to negative infinity.

  1. For the integral from ( 0 ) to ( \infty ):

[ \int_{0}^{\infty} x^3 , dx ]

Apply the antiderivative:

[ \left[ \frac{x^4}{4} \right]_{0}^{\infty} ]

As ( x ) approaches infinity, ( x^4 ) also grows without bound, so this integral diverges to positive infinity.

Since both parts of the improper integral diverge, the entire improper integral ( \int_{-\infty}^{\infty} x^3 , dx ) diverges.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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