What is the integral of ((a^x-b^x))/(a^xb^x) ?

The integral of ((a^x-b^x))/(a^xb^x) is -(b^{-x})/(ln(b))+(a^{-x})/(ln(a))+C

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integral of ((a^x-b^x))/(a^xb^x)

\int \frac{( a ^{x} - b ^{x} )}{a ^{x} b ^{x}} d x

- \frac{b ^{- x}}{ln ( b )} + \frac{a ^{- x}}{ln ( a )} + C

Solution steps

\int \frac{a ^{x} - b ^{x}}{a ^{x} b ^{x}} d x

Expand

\frac{a ^{x} - b ^{x}}{a ^{x} b ^{x}} : \frac{1}{b ^{x}} - \frac{1}{a ^{x}}

= \int \frac{1}{b ^{x}} - \frac{1}{a ^{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 \frac{1}{b ^{x}} d x - \int \frac{1}{a ^{x}} d x

\int \frac{1}{b ^{x}} d x = - \frac{b ^{- x}}{ln ( b )}

\int \frac{1}{a ^{x}} d x = - \frac{a ^{- x}}{ln ( a )}

= - \frac{b ^{- x}}{ln ( b )} - (- \frac{a ^{- x}}{ln ( a )})

Simplify

= - \frac{b ^{- x}}{ln ( b )} + \frac{a ^{- x}}{ln ( a )}

Add a constant to the solution

= - \frac{b ^{- x}}{ln ( b )} + \frac{a ^{- x}}{ln ( a )} + C

What is the integral of ((a^x-b^x))/(a^xb^x) ?
The integral of ((a^x-b^x))/(a^xb^x) is -(b^{-x})/(ln(b))+(a^{-x})/(ln(a))+C