# How do you integrate #int lnx/x^7# by integration by parts method?

We have that

#int lnx*[-x^-6/6]'dx=-1/6*lnx*x^-6+1/6*int(lnx)' x^-6dx=
-1/6lnx*x^-6+1/6

*int 1/x*x^-6dx= -1/6

*lnx*x^-6+1/6

*int x^-7dx= -1/6*lnx

*x^-6+1/6*int (x^-6/6)'dx= -1/6

*lnx*x^-6-(x^-6/36)+c= -(6

*lnx+1)/(36*x^6)+c#

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To integrate (\int \frac{\ln(x)}{x^7} dx) by the integration by parts method, we use the formula (\int u , dv = uv - \int v , du), where one part is differentiated ((du)) and the other is integrated ((dv)).

Let:

- (u = \ln(x)), which implies (du = \frac{1}{x} dx)
- (dv = \frac{1}{x^7} dx), which implies (v = \int \frac{1}{x^7} dx = -\frac{1}{6x^6})

Now, apply the integration by parts formula:

[ \int \frac{\ln(x)}{x^7} dx = uv - \int v , du ]

Substituting (u), (du), (v), and (dv) gives:

[ \int \frac{\ln(x)}{x^7} dx = -\frac{\ln(x)}{6x^6} - \int -\frac{1}{6x^6} \cdot \frac{1}{x} dx ]

[ = -\frac{\ln(x)}{6x^6} + \frac{1}{6} \int \frac{1}{x^7} dx ]

The integral on the right can be simplified further:

[ \frac{1}{6} \int \frac{1}{x^7} dx = \frac{1}{6} \left(-\frac{1}{6x^6}\right) = -\frac{1}{36x^6} ]

So, the complete solution is:

[ \int \frac{\ln(x)}{x^7} dx = -\frac{\ln(x)}{6x^6} - \frac{1}{36x^6} + C ]

where (C) is the constant of integration.

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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.

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