What is the integral of (a^x+b^x)/(c^x) ?

The integral of (a^x+b^x)/(c^x) is (a^x)/(c^xln(a/c))+(b^x)/(c^xln(b/c))+C

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integral of (a^x+b^x)/(c^x)

\int \frac{a ^{x} + b ^{x}}{c ^{x}} d x

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

Solution steps

\int \frac{a ^{x} + b ^{x}}{c ^{x}} d x

Expand

\frac{a ^{x} + b ^{x}}{c ^{x}} : \frac{a ^{x}}{c ^{x}} + \frac{b ^{x}}{c ^{x}}

= \int \frac{a ^{x}}{c ^{x}} + \frac{b ^{x}}{c ^{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{a ^{x}}{c ^{x}} d x + \int \frac{b ^{x}}{c ^{x}} d x

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

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

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

Add a constant to the solution

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

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What is the integral of (a^x+b^x)/(c^x) ?
The integral of (a^x+b^x)/(c^x) is (a^x)/(c^xln(a/c))+(b^x)/(c^xln(b/c))+C