What is the integral of ae^{-mx} ?

The integral of ae^{-mx} is -ae^{-mx} 1/m+C

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integral of ae^{-mx}

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

- a e^{- m x} \frac{1}{m} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{e ^{u}}{m} d u

= a \cdot \int - \frac{e ^{u}}{m} d u

Take the constant out:

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

= a (- \frac{1}{m} \cdot \int e^{u} d u)

Use the common integral:

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

= a (- \frac{1}{m} e^{u})

Substitute back

u = - m x

= a (- \frac{1}{m} e^{- m x})

Simplify

= - a e^{- m x} \frac{1}{m}

Add a constant to the solution

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

\frac{d y}{d x} + (\frac{3}{x}) y = - 2 x + 7

x \to - 1 lim (1 + 2 x)

100 y^{^{''}} + 60 y^{^{'}} - 27 y = 0

7 x, \frac{7}{4} x, 78 - x^{2}, y = 0