# Question Video: Finding the Integration of a Function Involving Logarithmic Function Using Integration by Substitution Mathematics • Higher Education

Determine β« β3/(π₯ ln 8π₯) dπ₯.

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### Video Transcript

Determine the indefinite integral of negative three over π₯ multiplied by the natural logarithm of eight π₯ with respect to π₯.

Now, although this may look like a tricky integral to evaluate, it is in fact in a form which we know how to integrate. If we let π of π₯ be equal to the natural logarithm of eight π₯, then we can differentiate π of π₯ using the fact that the differential of the natural logarithm of π₯ is one over π₯ in order to find that π prime of π₯ is equal to one over eight π₯. And then, since this is a composite function, we have eight π₯ inside the function of the natural logarithm. We mustnβt forget to multiply by the differential of eight π₯, which is just eight. This is because of the chain rule. Simplifying, we can obtain that π prime of π₯ is equal to one over π₯.

Now letβs rewrite our integral. If we multiply the numerator and denominator of our fraction by one over π₯, then we can rewrite our integral as the integral of negative three over π₯ over the natural logarithm of eight π₯ with respect to π₯. And now we can factor the negative three in the numerator. And once weβve reached this stage, we notice that this is in a form which we know how to integrate. Since itβs of the form the integral of π multiplied by π prime of π₯ over π of π₯ dπ₯. Where our π of π₯ is the natural logarithm of eight π₯. And our π prime of π₯ is one over π₯. Therefore, our value of π is negative three. Now we know that this integral evaluates to π multiplied by the natural logarithm of the absolute value of π of π₯ plus π. And we can simply substitute in the values of π and π of π₯ to find our solution. Which is negative three multiplied by the natural logarithm of the absolute value of the natural logarithm of eight π₯ plus π.