What is the integral of 4/(e^{-4x)} ?

The integral of 4/(e^{-4x)} is e^{4x}+C

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integral of 4/(e^{-4x)}

\int \frac{4}{e ^{- 4 x}} d x

e^{4 x} + C

Solution steps

\int \frac{4}{e ^{- 4 x}} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{e ^{- 4 x}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{e ^{- 4 x}} = e^{4 x}

= 4 \cdot \int e^{4 x} d x

Apply u-substitution

= 4 \cdot \int e^{u} \frac{1}{4} d u

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \int e^{u} d u

Use the common integral:

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

= 4 \cdot \frac{1}{4} e^{u}

Substitute back

u = 4 x

= 4 \cdot \frac{1}{4} e^{4 x}

Simplify

4 \cdot \frac{1}{4} e^{4 x} : e^{4 x}

= e^{4 x}

Add a constant to the solution

= e^{4 x} + C