How do you test the improper integral #int (2x-1)^3 dx# from #(-oo, oo)# and evaluate if possible?

Answer 1

Diverges.

First, let's obtain the indefinite integral #int(2x-1)^3dx# so as to make our work easier when evaluating the improper integral:
#u=2x-1, du=2dx, 1/2du=dx#
#1/2intu^3du=1/8u^4=1/8(2x-3)^4#

No need to include a constant of integration, this was only calculated to keep it on the side for when it becomes needed for evaluating the improper integral.

So,

#int_-oo^oo(2x-3)^3dx=int_-oo^0(2x-3)^3dx+int_0^oo(2x-3)^3dx#
This sort of splitting up is necessary when evaluating from #(-oo, oo).# It's often easier to split up at zero (unless zero is not in the domain of the integrand).
#=lim_(t->-oo)int_t^0(2x-3)^3dx+lim_(t->oo)int_0^t(2x-3)dx#
#=lim_(t->-oo)(1/8(2x-3)^4)|_t^0 + lim_(t->oo)(1/8(2x-3)^4)|_0^t#
#=lim_(t->-oo)(1/8(-3)^4-1/8(2t-3)^4)+lim_(t->oo)(1/8(2t-3)^4-1/8(-3)^4)=-81/8-oo+...#

The first limit did not exist, it went to negative infinity. No need to even evaluate the second one, we already know the entire integral must diverge.

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Answer 2

To test the improper integral ( \int_{-\infty}^{\infty} (2x - 1)^3 , dx ) for convergence and evaluate it if possible, you can follow these steps:

  1. Split the integral into two separate integrals: [ \int_{-\infty}^{\infty} (2x - 1)^3 , dx = \int_{-\infty}^{0} (2x - 1)^3 , dx + \int_{0}^{\infty} (2x - 1)^3 , dx ]

  2. For each of these integrals, evaluate the limit as the bound approaches infinity: [ \lim_{a \to -\infty} \int_{a}^{0} (2x - 1)^3 , dx ] and [ \lim_{b \to \infty} \int_{0}^{b} (2x - 1)^3 , dx ]

  3. Check if both limits exist and are finite. If both limits exist and are finite, the original integral converges. Otherwise, it diverges.

  4. If the integral converges, evaluate each integral separately using standard integration techniques. Otherwise, it cannot be evaluated.

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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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