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$$ \int 2^{3x} \times 5^x \times 3^{2x} dx $$

I think we're supposed to convert all the terms into log form, but I'm not sure, and other than that I have no idea how to tackle this problem.

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4 Answers 4

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hint: $$2^{3x}\cdot 5^x\cdot 3^{2x} = (2^3\cdot 5\cdot 3^2)^x$$

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$2^{3x} = (2^3)^x = 8^x$ and $3^{2x} = (3^2)^x = 9^x$ and integration of $a^x = \frac{a^x}{\ln(a)}$ and so your question is very easy now

$$ \int 2^{3x} \times 5^x \times 3^{2x} = \int 8^x \times 5^x \times 9^x$$ which is equal to $\int (8 \times 5 \times 9)^x$ which is $\int (360)^x$ and so the result is $$\frac{360^x}{\ln{360}}$$

Notice that I used the property that $$\color{blue}{(abc)^x = a^xb^xc^x}$$ where $a,b,c,x$ are integers

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The integrand is $360^x$. Does that help?

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Hint:

$$c^x = e^{\log ( c ) x}.$$

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