How do you evaluate the integral of #int x^3 lnx dx#?

Answer 1

#intx^3lnxdx=(x^4(4lnx-1))/16+C#

Use integration by parts, which states that:

#intudv=uv-intvdu#
So, for #intx^3lnxdx#, let #u=lnx# and #dv=x^3dx#.
These imply that #du=1/xdx# and #v=x^4/4# (obtain these by differentiating #u# and integrating #dv#, respectively).

Plugging these into the integration by parts formula, this yields:

#intx^3lnxdx=lnx(x^4/4)-int(x^4/4)(1/x)dx#
#=(x^4lnx)/4-1/4intx^3dx#
#=(x^4lnx)/4-x^4/16+C#
#=(x^4(4lnx-1))/16+C#
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Answer 2

To evaluate the integral ∫x^3 ln(x) dx, you can use integration by parts. Let u = ln(x) and dv = x^3 dx.

Differentiate u to find du, and integrate dv to find v. Then, apply the integration by parts formula:

∫u dv = uv - ∫v du

Now, differentiate u = ln(x) to find du:

du = (1/x) dx

Integrate dv = x^3 dx to find v:

v = (1/4)x^4

Now, apply the integration by parts formula:

∫x^3 ln(x) dx = uv - ∫v du = (ln(x))(1/4)x^4 - ∫(1/4)x^4 (1/x) dx = (1/4)x^4 ln(x) - (1/4) ∫x^3 dx

Now, integrate ∫x^3 dx:

= (1/4)x^4 ln(x) - (1/4) * (1/4)x^4 + C = (1/4)x^4 ln(x) - (1/16)x^4 + C

So, the integral of ∫x^3 ln(x) dx is:

(1/4)x^4 ln(x) - (1/16)x^4 + C, where C is the constant of integration.

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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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