What is the integral of e^{-19x} ?

The integral of e^{-19x} is -1/19 e^{-19x}+C

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integral of e^{-19x}

\int e^{- 19 x} d x

- \frac{1}{19} e^{- 19 x} + C

Solution steps

\int e^{- 19 x} d x

Apply u-substitution

: \int - \frac{1}{19} e^{u} d u

= \int - \frac{1}{19} e^{u} d u

Take the constant out:

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

= - \frac{1}{19} \cdot \int e^{u} d u

Use the common integral:

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

= - \frac{1}{19} e^{u}

Substitute back

u = - 19 x

= - \frac{1}{19} e^{- 19 x}

Add a constant to the solution

= - \frac{1}{19} e^{- 19 x} + C

- 3 x - x^{2}, - 10

y^{^{'}} = 9 x + y, y (0) = 3

y = x^{2} 9 - x^{2}

x \to 0 lim (\frac{7 x}{sin ( x )})

f (x) = 5 x + 6