Author Topic: how do i integrate this exponential function (Read 958 times) Tweet Share

nastya0987 · « **on:** February 26, 2017, 09:33:22 pm »

(1/3) * (3/4)^x

where * = multiply

thanks in advance

RuiAce · « **Reply #1 on:** February 26, 2017, 10:09:34 pm »

jakesilove · « **Reply #2 on:** February 26, 2017, 10:22:28 pm »

Quote from: RuiAce on February 26, 2017, 10:09:34 pm

$\text{Using the fact that }x=e^{\ln x}\text{ for all }x > 0\text{ we have}$
$\begin{align*}\frac13\int \left(\frac34\right)^xdx&=\frac13\int e^{\ln\left(\frac34\right)^x }dx\\ &= \frac13 \int e^{x \ln\left(\frac34\right)}dx\tag{log law}\end{align*}\\ \text{which is a standard integral and integrates into}\\ \frac13 \times \frac{e^{x\ln \frac34}}{\ln \frac34}\\ \text{so backtracking, becomes }\frac13 \times \frac{\left(\frac34\right)^x}{\ln \frac34}$

I was just gonna cite this off as a rule; Rui, the better Mathematician, chose to prove it. You are a legend as always!

RuiAce · « **Reply #3 on:** February 26, 2017, 10:25:15 pm »

jamonwindeyer · « **Reply #4 on:** February 27, 2017, 12:24:07 am »

Quote from: RuiAce on February 26, 2017, 10:25:15 pm

$\text{I have doubts that 2/3U are allowed to quote it.}\\ \text{4U might be though, and if it's possible it's preferred.}$

I was always taught to derive it as it was needed, but then it virtually never appears in 2U/3U

with the reference sheet now there I think anything not on that sheet (and not elementary or a full version of a formula or something) should probably be derived to be safe, in exam scenarios at least

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Author Topic: how do i integrate this exponential function (Read 958 times) Tweet Share

nastya0987

how do i integrate this exponential function

RuiAce

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jakesilove

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RuiAce

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jamonwindeyer

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