What is the integral of e^x+e^{ax}+e^{ax+b} ?

The integral of e^x+e^{ax}+e^{ax+b} is e^x+1/a e^{ax}+1/a e^{ax+b}+C

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integral of e^x+e^{ax}+e^{ax+b}

\int e^{x} + e^{a x} + e^{a x + b} d x

e^{x} + \frac{1}{a} e^{a x} + \frac{1}{a} e^{a x + b} + C

Solution steps

\int e^{x} + e^{a x} + e^{a x + b} 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 e^{a x} d x + \int e^{a x + b} d x

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

\int e^{a x} d x = \frac{1}{a} e^{a x}

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

= e^{x} + \frac{1}{a} e^{a x} + \frac{1}{a} e^{a x + b}

Add a constant to the solution

= e^{x} + \frac{1}{a} e^{a x} + \frac{1}{a} e^{a x + b} + C

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What is the integral of e^x+e^{ax}+e^{ax+b} ?
The integral of e^x+e^{ax}+e^{ax+b} is e^x+1/a e^{ax}+1/a e^{ax+b}+C