What is the integral of 1/(e^{a*x)} ?

The integral of 1/(e^{a*x)} is x/(e^{ax)}-e^{-ax}x-(e^{-ax})/a+C

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integral of 1/(e^{a*x)}

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

\frac{x}{e ^{a x}} - e^{- a x} x - \frac{e ^{- a x}}{a} + C

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

= \frac{x}{e ^{a x}} - e^{- a x} x - \frac{e ^{- a x}}{a}

Add a constant to the solution

= \frac{x}{e ^{a x}} - e^{- a x} x - \frac{e ^{- a x}}{a} + C

What is the integral of 1/(e^{a*x)} ?
The integral of 1/(e^{a*x)} is x/(e^{ax)}-e^{-ax}x-(e^{-ax})/a+C