How do you integrate #int x(lnx)^2 # using integration by parts?
The answer is
The integration by parts is
Here,
Therefore, the integral is
Therefore,
Putting all together
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To integrate ( \int x(\ln x)^2 ) using integration by parts, you would use the formula:
[ \int u , dv = uv - \int v , du ]
Let ( u = (\ln x)^2 ) and ( dv = x , dx ).
Then, differentiate ( u ) to find ( du ) and integrate ( dv ) to find ( v ).
[ du = 2\ln x \cdot \frac{1}{x} , dx = \frac{2}{x} \ln x , dx ] [ v = \frac{x^2}{2} ]
Now, apply the integration by parts formula:
[ \int x(\ln x)^2 , dx = uv - \int v , du ] [ = \frac{x^2}{2} (\ln x)^2 - \int \frac{x^2}{2} \cdot \frac{2}{x} \ln x , dx ]
Simplify and integrate the remaining term:
[ = \frac{x^2}{2} (\ln x)^2 - \int x \ln x , dx ]
This remaining integral can be further simplified and evaluated using integration by parts or other methods if necessary.
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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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