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

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

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integral of 3e^{-2x}

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{1}{2} e^{u} d u

= 3 \cdot \int - \frac{1}{2} e^{u} d u

Take the constant out:

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

= 3 (- \frac{1}{2} \cdot \int e^{u} d u)

Use the common integral:

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

= 3 (- \frac{1}{2} e^{u})

Substitute back

u = - 2 x

= 3 (- \frac{1}{2} e^{- 2 x})

Simplify

3 (- \frac{1}{2} e^{- 2 x}) : - \frac{3}{2} e^{- 2 x}

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

Add a constant to the solution

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

\frac{2 x ^{4}}{cos ( x ) e ^{x}}

x \to 3 + lim (\frac{∣ x - 3 ∣}{x - 3})

\int \frac{1}{1 - sin ( x )} d x

x \to 0 lim (\frac{sin ( 2 ) ( x )}{x})

\frac{1}{u ^{2}}