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

The integral of (2*3^x)/(2^x) is (3^x*2^{1-x})/(ln(3/2))+C

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

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

\frac{3 ^{x} \cdot 2 ^{1 - x}}{ln ( \frac{3}{2} )} + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply exponent rule:

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

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

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

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

Simplify

2 \cdot \frac{( \frac{3}{2} ) ^{x}}{ln ( \frac{3}{2} )} : \frac{3 ^{x} \cdot 2 ^{1 - x}}{ln ( \frac{3}{2} )}

= \frac{3 ^{x} \cdot 2 ^{1 - x}}{ln ( \frac{3}{2} )}

Add a constant to the solution

= \frac{3 ^{x} \cdot 2 ^{1 - x}}{ln ( \frac{3}{2} )} + C

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What is the integral of (2*3^x)/(2^x) ?
The integral of (2*3^x)/(2^x) is (3^x*2^{1-x})/(ln(3/2))+C