How do you integrate #int x^2 ln ^2x dx # using integration by parts?

Answer 1

#int x^2ln^2xdx = x^3/27(9ln^2x - 6lnx +2) +C#

We can integrate by parts using the logarithm as integral part, so that in the resulting integral we have a rational function:

#int x^2ln^2xdx = int ln^2x d(x^3/3)#
#int x^2ln^2xdx = (x^3ln^2x)/3 - 1/3 int x^3d(ln^2x)#
#int x^2ln^2xdx = (x^3ln^2x)/3 - 2/3 int x^3 lnx/xdx #
#int x^2ln^2xdx = (x^3ln^2x)/3 - 2/3 int x^2lnxdx #

Solve the resulting integral by parts again:

#int x^2lnxdx = int lnx d(x^3/3)#
#int x^2lnxdx = (x^3lnx)/3 - 1/3 int x^3 d(lnx)#
#int x^2lnxdx = (x^3lnx)/3 - 1/3 int x^3 dx/x#
#int x^2lnxdx = (x^3lnx)/3 - 1/3 int x^2 dx#
#int x^2lnxdx = (x^3lnx)/3 - 1/9x^3 +C#

Substituting in the first expression:

#int x^2ln^2xdx = (x^3ln^2x)/3 - 2/3( (x^3lnx)/3 - 1/9x^3 ) +C#

and simplifying:

#int x^2ln^2xdx = (x^3ln^2x)/3 - 2/9(x^3lnx) + 2/27x^3 +C#
#int x^2ln^2xdx = x^3/27(9ln^2x - 6lnx +2) +C#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To integrate ( \int x^2 \ln^2(x) , dx ) using integration by parts, we choose ( u = \ln^2(x) ) and ( dv = x^2 , dx ). Then, we find ( du ) and ( v ) as follows:

[ du = \frac{d}{dx} (\ln^2(x)) , dx = 2\ln(x) \frac{1}{x} , dx = \frac{2\ln(x)}{x} , dx ]

[ v = \int x^2 , dx = \frac{x^3}{3} ]

Now, we apply the integration by parts formula:

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

[ \int x^2 \ln^2(x) , dx = \frac{x^3}{3} \ln^2(x) - \int \frac{2x^3 \ln(x)}{3x} , dx ]

[ = \frac{x^3}{3} \ln^2(x) - \frac{2}{3} \int x^2 \ln(x) , dx ]

Now, we can solve the remaining integral ( \int x^2 \ln(x) , dx ) using integration by parts again or by using other methods like substitution.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To integrate (x^2 \ln^2(x) , dx) using integration by parts, follow these steps:

  1. Identify the parts of the integrand to assign to (u) and (dv).
  2. Compute (du) and (v) by differentiating and integrating (u) and (dv), respectively.
  3. Apply the integration by parts formula: (\int u , dv = uv - \int v , du).
  4. Substitute the computed values into the formula and integrate.

Given (u = \ln^2(x)) and (dv = x^2 , dx), we can compute (du) and (v) as follows:

[ du = \frac{d}{dx}(\ln^2(x)) , dx = 2 \ln(x) \frac{1}{x} , dx = \frac{2}{x} \ln(x) , dx ]

[ v = \int x^2 , dx = \frac{x^3}{3} ]

Now, apply the integration by parts formula:

[ \int x^2 \ln^2(x) , dx = uv - \int v , du ]

[ = \frac{x^3}{3} \ln^2(x) - \int \frac{x^3}{3} \cdot \frac{2}{x} \ln(x) , dx ]

[ = \frac{x^3}{3} \ln^2(x) - \frac{2}{3} \int x^2 \ln(x) , dx ]

At this point, you may need to integrate (\int x^2 \ln(x) , dx) again using integration by parts or another method. However, it's worth noting that this integral does not have a simple closed-form solution and may involve special functions.

So, the final expression for (\int x^2 \ln^2(x) , dx) using integration by parts is:

[ \frac{x^3}{3} \ln^2(x) - \frac{2}{3} \int x^2 \ln(x) , dx ]

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7