Video: Finding the Integration of a Function Involving Logarithmic Function Using Integration by Substitution

Determine ∫ βˆ’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 𝑓 of π‘₯ be equal to the natural logarithm of eight π‘₯, then we can differentiate 𝑓 of π‘₯ using the fact that the differential of the natural logarithm of π‘₯ is one over π‘₯ in order to find that 𝑓 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 mustn’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 𝑓 prime of π‘₯ is equal to one over π‘₯.

Now let’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 we’ve reached this stage, we notice that this is in a form which we know how to integrate. Since it’s of the form the integral of π‘Ž multiplied by 𝑓 prime of π‘₯ over 𝑓 of π‘₯ dπ‘₯. Where our 𝑓 of π‘₯ is the natural logarithm of eight π‘₯. And our 𝑓 prime of π‘₯ is one over π‘₯. Therefore, our value of π‘Ž is negative three. Now we know that this integral evaluates to π‘Ž multiplied by the natural logarithm of the absolute value of 𝑓 of π‘₯ plus 𝑐. And we can simply substitute in the values of π‘Ž and 𝑓 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 𝑐.

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