Joachim
  • Joachim
how would one integrate this function:
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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Joachim
  • Joachim
\[3^{(x(\ln(4x+5))/4}\]
Joachim
  • Joachim
and find the derivative...
.Sam.
  • .Sam.
I don't think you can integrate that

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.Sam.
  • .Sam.
For differentiation, there's whole lot of steps, Rough guess Power rule ---> Product rule ---> Chain rule ---> Product rule again ---> chain rule again
Joachim
  • Joachim
sorry, the integration was a typo - got that one mixed up with differentiation
.Sam.
  • .Sam.
\[\ \text{Use the chain rule}\] \[\frac{d}{dx}\left(3^{\frac{1}{4} x \log (4 x+5)}\right)=\frac{d3^u}{du} \frac{du}{dx}\] \[\text{where }u=\frac{1}{4} x \log (4 x+5)\text{ and }\frac{d3^u}{du}=3^u \log (3)\] \[\log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\frac{d}{dx}\left(\frac{1}{4} x \log (4 x+5)\right)\right)\] \[\text{Use the product rule, }\frac{d}{dx}(u v)=v \frac{du}{dx}+u \frac{dv}{dx}\] \[\text{where }u=x\text{ and }v=\log (4 x+5)\] \[\frac{1}{4} \log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\log (4 x+5) \left(\frac{d}{dx}(x)\right)+x \left(\frac{d}{dx}(\log (4 x+5))\right)\right)\] \[\text{Use the chain rule AGAIN, }\frac{d}{dx}(\log (4 x+5))=\frac{d\log (u)}{du} \frac{du}{dx}\] \[\text{where }u=4 x+5\text{ and }\frac{d\log (u)}{du}=\frac{1}{u}\frac{1}{4} \log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(x \frac{\frac{d}{dx}(4 x+5)}{4 x+5}+\log (4 x+5) \left(\frac{d}{dx}(x)\right)\right)\] \[\frac{1}{4} \log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\frac{x \left(4 \left(\frac{d}{dx}(x)\right)+\frac{d}{dx}(5)\right)}{4 x+5}+\log (4 x+5) \left(\frac{d}{dx}(x)\right)\right)\] Simplify the derivatives, \[\frac{1}{4} \log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\frac{4 x}{4 x+5}+\log (4 x+5)\right)\] \[\log (3) 3^{\frac{1}{4} x \log (4 x+5)} \left(\frac{x}{4 x+5}+\frac{1}{4} \log (4 x+5)\right)\]

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