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

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

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integral of (2^x)/(5^x)

\int \frac{2 ^{x}}{5 ^{x}} d x

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

Solution steps

\int \frac{2 ^{x}}{5 ^{x}} d x

Apply exponent rule:

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{3}{sin ( x ) + cos ( x )})

x \to 0 lim (cot (2 x))

\int (x - 5) sin (π x) d x

\frac{\partial}{\partial x} (- \frac{8 x}{y})

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