How do you integrate #int x^4(lnx)^2# by integration by parts method?
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To integrate ( \int x^4 (\ln x)^2 ) by integration by parts, we use the formula:
[ \int u , dv = uv  \int v , du ]
We choose ( u = \ln^2 x ) and ( dv = x^4 , dx ). Then, we find ( du ) and ( v ):
[ du = 2\ln x \frac{1}{x} , dx = \frac{2}{x} \ln x , dx ]
[ v = \frac{1}{5}x^5 ]
Now, we apply the integration by parts formula:
[ \int x^4 (\ln x)^2 , dx = \frac{1}{5}x^5 \ln^2 x  \int \frac{2}{x} \ln x \cdot \frac{1}{5}x^5 , dx ]
[ = \frac{1}{5}x^5 \ln^2 x  \frac{2}{5} \int x^4 \ln x , dx ]
Now, we use integration by parts again for the second integral. We choose ( u = \ln x ) and ( dv = x^4 , dx ). Then, we find ( du ) and ( v ):
[ du = \frac{1}{x} , dx ]
[ v = \frac{1}{5}x^5 ]
Applying the integration by parts formula again:
[ \int x^4 (\ln x)^2 , dx = \frac{1}{5}x^5 \ln^2 x  \frac{2}{5} \left( \frac{1}{5}x^5 \ln x  \int \frac{1}{5}x^4 \cdot \frac{1}{x} , dx \right) ]
[ = \frac{1}{5}x^5 \ln^2 x  \frac{2}{5} \left( \frac{1}{5}x^5 \ln x  \frac{1}{5} \int x^3 , dx \right) ]
[ = \frac{1}{5}x^5 \ln^2 x  \frac{2}{5} \left( \frac{1}{5}x^5 \ln x  \frac{1}{20}x^4 \right) + C ]
[ = \frac{1}{5}x^5 \ln^2 x  \frac{2}{25}x^5 \ln x + \frac{1}{50}x^4 + C ]
So, ( \int x^4 (\ln x)^2 , dx = \frac{1}{5}x^5 \ln^2 x  \frac{2}{25}x^5 \ln x + \frac{1}{50}x^4 + C ).
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To integrate ( \int x^4 (\ln x)^2 ) using integration by parts, follow these steps:

Choose u and dv: Let ( u = \ln^2 x ) and ( dv = x^4 , dx ).

Find du and v: Calculate the derivatives ( du ) and the antiderivative ( v ) of ( dv ):
 ( du = 2\ln x \cdot \frac{1}{x} , dx = \frac{2}{x} \ln x , dx )
 ( v = \frac{1}{5} x^5 )

Apply the Integration by Parts formula: The formula for integration by parts is: [ \int u , dv = uv  \int v , du ]

Plug in the values: Substitute ( u ), ( v ), ( du ), and ( dv ) into the formula: [ \int x^4 (\ln x)^2 , dx = \frac{1}{5} x^5 \cdot \ln^2 x  \int \frac{2}{x} \ln x \cdot \frac{1}{5} x^5 , dx ]

Simplify and integrate: [ = \frac{1}{5} x^5 \cdot \ln^2 x  \frac{2}{5} \int x^4 \ln x , dx ]

Repeat the process: You may need to apply integration by parts again to the remaining integral ( \int x^4 \ln x , dx ).

Evaluate the integral: Once you've simplified the integral using integration by parts, evaluate the resulting expression.

Check for convergence: Ensure that the integral converges and the result is valid within the given domain.
By following these steps, you can integrate ( \int x^4 (\ln x)^2 ) using the integration by parts method.
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