What is the integral of (2*3^x+5*2^x)/(3^x) ?

The integral of (2*3^x+5*2^x)/(3^x) is 2x+(5*2^x)/(ln(2/3)*3^x)+C

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integral of (23^x+52^x)/(3^x)

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

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

Solution steps

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

Expand

\frac{2 \cdot 3 ^{x} + 5 \cdot 2 ^{x}}{3 ^{x}} : 2 + \frac{5 \cdot 2 ^{x}}{3 ^{x}}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 2 d x = 2 x

\int \frac{5 \cdot 2 ^{x}}{3 ^{x}} d x = \frac{5 \cdot 2 ^{x}}{ln ( \frac{2}{3} ) \cdot 3 ^{x}}

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

Add a constant to the solution

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

What is the integral of (2*3^x+5*2^x)/(3^x) ?
The integral of (2*3^x+5*2^x)/(3^x) is 2x+(5*2^x)/(ln(2/3)*3^x)+C