What is the integral of e^{-θx} ?

The integral of e^{-θx} is -1/θ e^{-θx}+C

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integral of e^{-θx}

\int e^{- θ x} d x

- \frac{1}{θ} e^{- θ x} + C

Solution steps

\int e^{- θ x} d x

Apply u-substitution

: \int - \frac{e ^{u}}{θ} d u

= \int - \frac{e ^{u}}{θ} d u

Take the constant out:

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

= - \frac{1}{θ} \cdot \int e^{u} d u

Use the common integral:

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

= - \frac{1}{θ} e^{u}

Substitute back

u = - θ x

= - \frac{1}{θ} e^{- θ x}

Add a constant to the solution

= - \frac{1}{θ} e^{- θ x} + C

\frac{d}{d x} (x + y - z)

f (x) = e^{5 x}

\int 2 x^{3} 1 + x^{2} d x

\frac{d y}{d t} = \frac{1}{( y + 2 ) ^{2}}, y (0) = 1

y = \frac{1}{1 + x ^{2}}, (1, \frac{1}{2})