How do you integrate #(x^5)*(e^(x^2/2))#?

Answer 1

#(4x^4-4x^2-8)e^(x^2/2)+c#

Let's assume #x^2/2 = t# #=> (2xdx)/2 = t, x dx = dt#
#=int(x^5)*(e^(x^2/2))dx# can be written as,
#=int(x^2)(x^2)x(e^(x^2/2))dx#
Substituting with #t# we get
#=int(2t)(2t)(e^t)dt#
#=4intt^2(e^t)dt#

Using Integration by Parts,

#∫(I)(II)dx=(I)∫(II)dx−∫((I)'∫(II)dx)dx#
where #(I)# and #(II)# are functions of #x#, and #(I)# represents which will be differentiated and #(II)# will be integrated subsequently in the above formula

Similarly following for the problem,

#=4(t^2)inte^tdt-4int((t^2)'inte^tdt)dt+c#
#=4t^2e^t-4int2te^tdt+c#
#=4t^2e^t-8intte^tdt+c#

Applying again integration by parts in second term, we get

#=4t^2e^t-8(te^t-inte^tdt)+c#
#=4t^2e^t-8(te^t-e^t)+c#
#=4t^2e^t-8te^t-8e^t+c#
Substituting #t# back,
#=4(x^2/2)^2e^(x^2/2)-8(x^2/2)e^(x^2/2)-8e^(x^2/2)+c#, where #c# is a constant
#=(4x^4-4x^2-8)e^(x^2/2)+c#
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Answer 2

To integrate (x^5 \cdot e^{\frac{x^2}{2}}), you can use integration by parts. Let (u = x^5) and (dv = e^{\frac{x^2}{2}} dx). Then, (du = 5x^4 dx) and (v = \int e^{\frac{x^2}{2}} dx). Integrating (v), we find (v = \sqrt{2\pi} \cdot \text{erf}\left(\frac{x}{\sqrt{2}}\right)), where (\text{erf}) denotes the error function.

Applying the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

We get:

[ \int x^5 \cdot e^{\frac{x^2}{2}} dx = x^5 \cdot \sqrt{2\pi} \cdot \text{erf}\left(\frac{x}{\sqrt{2}}\right) - 5 \int x^4 \cdot \sqrt{2\pi} \cdot \text{erf}\left(\frac{x}{\sqrt{2}}\right) dx ]

This expression still needs to be integrated, but now it's reduced to a simpler integral.

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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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