What is the integral of ((e^{8x}+1))/(e^{7x)} ?

The integral of ((e^{8x}+1))/(e^{7x)} is e^x-1/7 e^{-7x}+C

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integral of ((e^{8x}+1))/(e^{7x)}

\int \frac{( e ^{8 x} + 1 )}{e ^{7 x}} d x

e^{x} - \frac{1}{7} e^{- 7 x} + C

Solution steps

\int \frac{e ^{8 x} + 1}{e ^{7 x}} d x

Expand

\frac{e ^{8 x} + 1}{e ^{7 x}} : e^{x} + \frac{1}{e ^{7 x}}

= \int e^{x} + \frac{1}{e ^{7 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 e^{x} d x + \int \frac{1}{e ^{7 x}} d x

\int e^{x} d x = e^{x}

\int \frac{1}{e ^{7 x}} d x = - \frac{1}{7} e^{- 7 x}

= e^{x} - \frac{1}{7} e^{- 7 x}

Add a constant to the solution

= e^{x} - \frac{1}{7} e^{- 7 x} + C

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What is the integral of ((e^{8x}+1))/(e^{7x)} ?
The integral of ((e^{8x}+1))/(e^{7x)} is e^x-1/7 e^{-7x}+C