What is the integral of (e^x)/((1+x)) ?

The integral of (e^x)/((1+x)) is 1/e \Ei(1+x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (e^x)/((1+x))

integral

\frac{e ^{x}}{( 1 + x )}

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

Solution steps

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

Apply u-substitution

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

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

Simplify

e^{u - 1} : e^{u} e^{- 1}

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

Take the constant out:

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

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

Apply exponent rule:

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

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

Use the nonelementary integral:

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

= \frac{1}{e} Ei (u)

Substitute back

u = 1 + x

= \frac{1}{e} Ei (1 + x)

Add a constant to the solution

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

e^{4} x \cdot tan^{2} (x)

y = \frac{x}{2 + 1}

\frac{x}{( 1 + x - x ^{2} )}

(x + (x^{2} + 1)^{\frac{1}{2}})^{- 1}

\frac{1 + x}{1 - x}

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