WEBVTT
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Determine the integral of nine to the power of the logarithm base nine of π₯ with respect to π₯.
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Weβre asked to evaluate an integral.
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And we can see that our integrand is in a special form.
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We need to recall what the logarithm base π of π₯ actually means.
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Itβs defined to be the inverse function of π to the power of π₯.
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And if these two functions are inverse of one another, π to the power of the logarithm base π of π₯ will just be equal to π₯.
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Because theyβre inverse functions, we just have the identity function.
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We donβt change the value of our input.
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So if we set π equal to nine, we see nine to the power of the log base nine of π₯ is just equal to π₯.
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So by using our laws of logarithms, weβve shown the integral given to us in the question is equal to the integral of π₯ with respect to π₯.
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And we can evaluate this by using the power rule for integration.
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We get π₯ squared over two plus πΆ.
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Therefore, weβve shown the integral of nine to the power of log base nine of π₯ with respect to π₯ is equal to π₯ squared over two plus πΆ.