# How do you integrate #(5 ln(x))/x^(7)#?

The answer is

Apply the Integration by parts

The integral is

Therefore,

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To integrate ( \frac{5 \ln(x)}{x^7} ), you can use integration by parts. Let ( u = \ln(x) ) and ( dv = \frac{5}{x^7}dx ). Then, ( du = \frac{1}{x}dx ) and ( v = -\frac{1}{6x^6} ).

Applying the integration by parts formula:

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

Substituting the values:

[ \int \frac{5 \ln(x)}{x^7} , dx = -\frac{5 \ln(x)}{6x^6} - \int \left(-\frac{1}{6x^6}\right) \cdot \frac{1}{x} , dx ]

Simplify the integral:

[ \int \frac{5 \ln(x)}{x^7} , dx = -\frac{5 \ln(x)}{6x^6} + \frac{1}{6} \int \frac{1}{x^7} , dx ]

Now, integrate ( \frac{1}{x^7} ):

[ \int \frac{1}{x^7} , dx = \frac{-1}{6x^6} + C ]

So, putting it all together:

[ \int \frac{5 \ln(x)}{x^7} , dx = -\frac{5 \ln(x)}{6x^6} - \frac{1}{6} \left(\frac{-1}{6x^6}\right) + C ]

[ = -\frac{5 \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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