What is the integral of (2^x)/(3^x) ?

The integral of (2^x)/(3^x) is (2^x)/(ln(2/3)*3^x)+C

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integral of (2^x)/(3^x)

\int \frac{2 ^{x}}{3 ^{x}} d x

\frac{2 ^{x}}{ln ( \frac{2}{3} ) \cdot 3 ^{x}} + C

Solution steps

\int \frac{2 ^{x}}{3 ^{x}} d x

Apply exponent rule:

\frac{a ^{c}}{b ^{c}} = (\frac{a}{b})^{c}

= \int (\frac{2}{3})^{x} d x

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{( \frac{2}{3} ) ^{x}}{ln ( \frac{2}{3} )}

Simplify

\frac{( \frac{2}{3} ) ^{x}}{ln ( \frac{2}{3} )} : \frac{2 ^{x}}{ln ( \frac{2}{3} ) \cdot 3 ^{x}}

= \frac{2 ^{x}}{ln ( \frac{2}{3} ) \cdot 3 ^{x}}

Add a constant to the solution

= \frac{2 ^{x}}{ln ( \frac{2}{3} ) \cdot 3 ^{x}} + C

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What is the integral of (2^x)/(3^x) ?
The integral of (2^x)/(3^x) is (2^x)/(ln(2/3)*3^x)+C