What is the integral of m/(e^{ax)} ?

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

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integral of m/(e^{ax)}

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

= m (\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}

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

Simplify

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

Add a constant to the solution

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

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