What is the integral of (2^x3^{x+1})/(5^{x+2)} ?

The integral of (2^x3^{x+1})/(5^{x+2)} is (2^x*3^{1+x})/(ln(6/5)*5^{2+x)}+C

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integral of (2^x3^{x+1})/(5^{x+2)}

\int \frac{2 ^{x} 3 ^{x + 1}}{5 ^{x + 2}} d x

\frac{2 ^{x} \cdot 3 ^{1 + x}}{ln ( \frac{6}{5} ) \cdot 5 ^{2 + x}} + C

Solution steps

\int \frac{2 ^{x} \cdot 3 ^{x + 1}}{5 ^{x + 2}} d x

\frac{2 ^{x} \cdot 3 ^{x + 1}}{5 ^{x + 2}} = \frac{3 ^{1}}{5 ^{2}} \frac{2 ^{x} \cdot 3 ^{x}}{5 ^{x}}

= \int \frac{3 ^{1}}{5 ^{2}} \cdot \frac{2 ^{x} \cdot 3 ^{x}}{5 ^{x}} d x

Take the constant out:

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

= \frac{3 ^{1}}{5 ^{2}} \cdot \int \frac{2 ^{x} \cdot 3 ^{x}}{5 ^{x}} d x

Simplify

= \frac{3}{25} \cdot \int \frac{2 ^{x} \cdot 3 ^{x}}{5 ^{x}} d x

Simplify

\frac{2 ^{x} \cdot 3 ^{x}}{5 ^{x}} : (\frac{6}{5})^{x}

= \frac{3}{25} \cdot \int (\frac{6}{5})^{x} d x

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

= \frac{3}{25} \cdot \frac{( \frac{6}{5} ) ^{x}}{ln ( \frac{6}{5} )}

Simplify

\frac{3}{25} \cdot \frac{( \frac{6}{5} ) ^{x}}{ln ( \frac{6}{5} )} : \frac{2 ^{x} \cdot 3 ^{1 + x}}{ln ( \frac{6}{5} ) \cdot 5 ^{2 + x}}

= \frac{2 ^{x} \cdot 3 ^{1 + x}}{ln ( \frac{6}{5} ) \cdot 5 ^{2 + x}}

Add a constant to the solution

= \frac{2 ^{x} \cdot 3 ^{1 + x}}{ln ( \frac{6}{5} ) \cdot 5 ^{2 + x}} + C

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What is the integral of (2^x3^{x+1})/(5^{x+2)} ?
The integral of (2^x3^{x+1})/(5^{x+2)} is (2^x*3^{1+x})/(ln(6/5)*5^{2+x)}+C