How do you find the Improper integral #int (x^2)e^[(-x^2)/2] dx # from x=-∞ to x=∞?

Answer 1

#= sqrt (2 pi)#

These two methods require external/prior knowledge, which is highlighted below

METHOD 1 let # I = int_{-oo}^{oo} x^2 \ e^[(- x^2)/2] dx#
# = 2 int_{0}^{oo} x^2 \ e^[(- x^2)/2] dx# as its symmetrical
#t = x^2/2, dt = x dx = sqrt(2t) \ dx#
# I = 2 int_{0}^{oo} 2t \ e^[- t] 1/sqrt(2t) \ dt#
# I =2 sqrt2 int_{0}^{oo} sqrt(t) \ e^[- t] \ dt qquad triangle#
PRIOR KNOWLEDGE: #mathcal(L) { sqrt(t)}(s) = 1/(2s) sqrt(pi/s)#
and #triangle# is simply the Laplace transform of #sqrt(t)# with s = 1
So #I =2 sqrt2* 1/2 sqrt pi = sqrt (2 pi)#

METHOD 2

PRIOR KNOWLEDGE: #int_(-oo)^(oo) \ e^(-alpha x^2) = sqrt(pi/alpha)#
differentiate wrt #alpha#
#d/(d alpha) int_(-oo)^(oo) \ e^(-alpha x^2) = d/(d alpha) sqrt(pi/alpha)#
# int_(-oo)^(oo) -x^2 \ e^(-alpha x^2) = -1/2 sqrt(pi) alpha^(-3/2) #
# int_(-oo)^(oo) x^2 \ e^(-alpha x^2) = 1/2 sqrt(pi/ alpha^3)#
#alpha = 1/2#
# int_(-oo)^(oo) x^2 \ e^(-x^2/2) = 1/2 sqrt(pi/ (1/2)^3) = sqrt (2 pi)#

that's the same answer with two slightly different approaches.

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Answer 2

To find the improper integral ∫(x^2)e^(-x^2/2) dx from x=-∞ to x=∞, we can use the properties of even functions and integrate over the entire real line. Since the integrand is even, we have:

∫(-∞ to ∞) (x^2)e^(-x^2/2) dx = 2 * ∫(0 to ∞) (x^2)e^(-x^2/2) dx

We can then use integration by parts to evaluate this integral. Let u = x^2 and dv = e^(-x^2/2) dx. Then, du = 2x dx and v = -e^(-x^2/2).

Applying the integration by parts formula:

∫ u dv = uv - ∫ v du

We get:

2 * ∫(0 to ∞) (x^2)e^(-x^2/2) dx = -2x^2 e^(-x^2/2) ∣(0 to ∞) + 4 * ∫(0 to ∞) x e^(-x^2/2) dx

Since e^(-x^2/2) approaches 0 as x approaches ∞, the first term in the integral evaluates to 0. Now, we can use a substitution, letting z = -x^2/2, so -dz = x dx:

4 * ∫(0 to ∞) x e^(-x^2/2) dx = -4 ∫(0 to ∞) e^z dz = -4e^z ∣(0 to ∞) = -4(0 - e^0) = -4(-1) = 4

Therefore, the improper integral ∫(-∞ to ∞) (x^2)e^(-x^2/2) dx converges and its value is 4.

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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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