What is the integral of 10^x(2^{-x}+5^{-x}) ?

The integral of 10^x(2^{-x}+5^{-x}) is (5^x)/(ln(5))+(2^x)/(ln(2))+C

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integral of 10^x(2^{-x}+5^{-x})

\int 1 0^{x} (2^{- x} + 5^{- x}) d x

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

Solution steps

\int 1 0^{x} (2^{- x} + 5^{- x}) d x

Expand

1 0^{x} (2^{- x} + 5^{- x}) : 5^{x} + 2^{x}

= \int 5^{x} + 2^{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 5^{x} d x + \int 2^{x} d x

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

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

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

Add a constant to the solution

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

\frac{d}{d x} (A x^{2} e^{2 x})

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

\frac{3}{2 s + 1} \cdot \frac{1}{s}

\frac{d}{d x} ((x - \frac{1}{x})^{2})

t \to \frac{x}{2} lim (\frac{sin ( t )}{t})

What is the integral of 10^x(2^{-x}+5^{-x}) ?
The integral of 10^x(2^{-x}+5^{-x}) is (5^x)/(ln(5))+(2^x)/(ln(2))+C