What is the integral of ((a^x-b^x)^2)/(a^x-b^x) ?

The integral of ((a^x-b^x)^2)/(a^x-b^x) is (a^x)/(ln(a))-(b^x)/(ln(b))+C

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integral of ((a^x-b^x)^2)/(a^x-b^x)

\int \frac{( a ^{x} - b ^{x} ) ^{2}}{a ^{x} - b ^{x}} d x

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

Solution steps

\int \frac{( a ^{x} - b ^{x} ) ^{2}}{a ^{x} - b ^{x}} d x

Cancel the common factor:

a^{x} - b^{x}

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

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

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

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

Add a constant to the solution

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

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What is the integral of ((a^x-b^x)^2)/(a^x-b^x) ?
The integral of ((a^x-b^x)^2)/(a^x-b^x) is (a^x)/(ln(a))-(b^x)/(ln(b))+C