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

The integral of (-e^{4x})/(4x) is -1/4 \Ei(4x)+C

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

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

- \frac{1}{4} Ei (4 x) + C

Solution steps

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

Take the constant out:

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

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

Simplify

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

Apply u-substitution

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

Use the nonelementary integral:

\int \frac{e ^{x}}{x} d x = Ei (x)

= (- \frac{1}{4}) Ei (u)

Substitute back

u = 4 x

= (- \frac{1}{4}) Ei (4 x)

Simplify

= - \frac{1}{4} Ei (4 x)

Add a constant to the solution

= - \frac{1}{4} Ei (4 x) + C

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What is the integral of (-e^{4x})/(4x) ?
The integral of (-e^{4x})/(4x) is -1/4 \Ei(4x)+C