What is the integral of e^{1/(3x)} ?

The integral of e^{1/(3x)} is e^{1/(3x)}x-1/3 \Ei(1/(3x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^{1/(3x)}

\int e^{\frac{1}{3 x}} d x

e^{\frac{1}{3 x}} x - \frac{1}{3} Ei (\frac{1}{3 x}) + C

Solution steps

\int e^{\frac{1}{3 x}} d x

Apply u-substitution

: \int e^{\frac{1}{u}} \frac{1}{3} d u

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

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{e ^{v}}{v ^{2}} d v

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

Take the constant out:

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

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

Apply Integration By Parts

: - \frac{e ^{v}}{v} - \int - \frac{e ^{v}}{v} d v

= \frac{1}{3} (- (- \frac{e ^{v}}{v} - \int - \frac{e ^{v}}{v} d v))

Take the constant out:

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

= \frac{1}{3} (- (- \frac{e ^{v}}{v} - (- \int \frac{e ^{v}}{v} d v)))

Use the nonelementary integral:

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

= \frac{1}{3} (- (- \frac{e ^{v}}{v} - (- Ei (v))))

Substitute back

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

Simplify

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

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

Add a constant to the solution

= e^{\frac{1}{3 x}} x - \frac{1}{3} Ei (\frac{1}{3 x}) + C

\int x^{- \frac{17}{10}} d x

\int (x^{2} - 2 x) cos (x) d x

\int \frac{t}{( t ^{2} + 1 ) ^{2}} d t

\int \frac{1}{sin ( 5 x )} d x

\int \frac{x}{( x - 1 ) ^{2} ( x - 2 )} d x

What is the integral of e^{1/(3x)} ?
The integral of e^{1/(3x)} is e^{1/(3x)}x-1/3 \Ei(1/(3x))+C