What is the integral of e^{-3/2 x} ?

The integral of e^{-3/2 x} is -2/3 e^{-(3x)/2}+C

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integral of e^{-3/2 x}

\int e^{- \frac{3}{2} x} d x

- \frac{2}{3} e^{- \frac{3 x}{2}} + C

Solution steps

\int e^{- \frac{3}{2} x} d x

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \int e^{- \frac{3 x}{2}} d x

Apply u-substitution

= \int - \frac{2}{3} e^{u} d u

Take the constant out:

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

= - \frac{2}{3} \cdot \int e^{u} d u

Use the common integral:

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

= - \frac{2}{3} e^{u}

Substitute back

u = - \frac{3 x}{2}

= - \frac{2}{3} e^{- \frac{3 x}{2}}

Add a constant to the solution

= - \frac{2}{3} e^{- \frac{3 x}{2}} + C

What is the integral of e^{-3/2 x} ?
The integral of e^{-3/2 x} is -2/3 e^{-(3x)/2}+C