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The integral between two and β of π to the negative five π with respect to π is convergent.
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What does it converge to?
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Because this integral has an infinite limit, we call this an improper integral.
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We have a general result that tells us that the integral between π and β of π of π₯ with respect to π₯ is the limit as π‘ approaches β of the integral between π and π‘ of π of π₯ with respect to π₯.
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And we know that this integral is convergent, which means the integral must approach a specific value.
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And thatβs what weβre going to find.
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What this result tells us is that we can replace the infinite limit in our integral with a variable π‘ and then take the limit as π‘ approaches β of our integral.
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So weβre going to find the limit as π‘ approaches β of the integral between two and π‘ of π to the power of negative five π with respect to π.
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To do this, we recall the general rule that the integral of π to the ππ₯ power with respect to π₯ is equal to one over π multiplied by π to the ππ₯ power plus a constant of integration π.
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So the integral of π to the power of negative five π with respect to π is one over negative five π to the power of negative five π.
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Note that because we have limits of integration for this question, we donβt need to include a constant of integration.
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Letβs write this as a single faction, negative π to the power of negative five π over five.
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And now, letβs apply these limits.
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And when we do this, we have to remember that our integral is negative.
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So when we apply the limits, weβre going to be subtracting a negative.
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So we can replace this with an add.
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And we can replace negative five multiplied by two with negative 10.
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But we still need to apply this limit.
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If we rewrite the term negative π to the power of negative five π‘ over five, using the fact that π to the negative five π‘ is the same as one over π to the five π‘, and then we recognize that we can simplify this to negative one over five π to the five π‘.
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And we know the fact that the limit as π‘ approaches β of π to the π‘ power is β.
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So the denominator approaches β.
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So the term approaches zero.
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So in fact, when we apply the limit to this bracket, what we have left is π to the negative 10 power over five, which shows that this limit approaches a specific value just as we expected.