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Find the integral $\int\frac{e^x}{x}dx$

Step-by-step Solution

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Solution

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Final Answer

$Ei\left(x\right)+C_0$
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Step-by-step Solution

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1

Rewrite the fraction $\frac{e^x}{x}$ inside the integral as the product of two functions: $e^x\frac{1}{x}$

$\int e^x\frac{1}{x}dx$
2

We can solve the integral $\int e^x\frac{1}{x}dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
3

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=e^x}\\ \displaystyle{du=e^xdx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\frac{1}{x}dx}\\ \displaystyle{\int dv=\int \frac{1}{x}dx}\end{matrix}$
5

Solve the integral

$v=\int\frac{1}{x}dx$
6

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\ln\left(x\right)$
7

Now replace the values of $u$, $du$ and $v$ in the last formula

$e^x\ln\left(x\right)-\int e^x\ln\left(x\right)dx$
8

We can solve the integral $\int e^x\ln\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
9

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=\ln\left(x\right)}\\ \displaystyle{du=\frac{1}{x}dx}\end{matrix}$
10

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=e^xdx}\\ \displaystyle{\int dv=\int e^xdx}\end{matrix}$
11

Solve the integral

$v=\int e^xdx$
12

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$e^x$
13

Now replace the values of $u$, $du$ and $v$ in the last formula

$e^x\ln\left(x\right)-\left(e^x\ln\left(x\right)-\int\frac{e^x}{x}dx\right)$
14

Multiply the single term $-1$ by each term of the polynomial $\left(e^x\ln\left(x\right)-\int\frac{e^x}{x}dx\right)$

$e^x\ln\left(x\right)-e^x\ln\left(x\right)+\int\frac{e^x}{x}dx$
15

Rewrite the fraction $\frac{e^x}{x}$ inside the integral as the product of two functions: $e^x\frac{1}{x}$

$e^x\ln\left(x\right)-e^x\ln\left(x\right)+\int e^x\frac{1}{x}dx$
16

We can solve the integral $\int e^x\frac{1}{x}dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
17

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=e^x}\\ \displaystyle{du=e^xdx}\end{matrix}$
18

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\frac{1}{x}dx}\\ \displaystyle{\int dv=\int \frac{1}{x}dx}\end{matrix}$
19

Solve the integral

$v=\int\frac{1}{x}dx$
20

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\ln\left(x\right)$
21

Now replace the values of $u$, $du$ and $v$ in the last formula

$e^x\ln\left(x\right)-e^x\ln\left(x\right)+e^x\ln\left(x\right)-\int e^x\ln\left(x\right)dx$
22

We can solve the integral $\int e^x\ln\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
23

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=\ln\left(x\right)}\\ \displaystyle{du=\frac{1}{x}dx}\end{matrix}$
24

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=e^xdx}\\ \displaystyle{\int dv=\int e^xdx}\end{matrix}$
25

Solve the integral

$v=\int e^xdx$
26

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$e^x$
27

Now replace the values of $u$, $du$ and $v$ in the last formula

$e^x\ln\left(x\right)-e^x\ln\left(x\right)+e^x\ln\left(x\right)-\left(e^x\ln\left(x\right)-\int\frac{e^x}{x}dx\right)$
28

Multiply the single term $-1$ by each term of the polynomial $\left(e^x\ln\left(x\right)-\int\frac{e^x}{x}dx\right)$

$e^x\ln\left(x\right)-e^x\ln\left(x\right)+e^x\ln\left(x\right)-e^x\ln\left(x\right)+\int\frac{e^x}{x}dx$
29

Simplify the expression inside the integral

$\int\frac{e^x}{x}dx$
30

The integral $\int\frac{e^x}{x}dx$ is called 'exponential integral' and is non-elementary. The formula for the exponential integral is: $\int\frac{e^x}{x}=Ei(x)$, where $Ei$ is a special function on the complex plane

$Ei\left(x\right)$
31

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$Ei\left(x\right)+C_0$

Final Answer

$Ei\left(x\right)+C_0$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve integral of ((e^x)/x)dx using partial fractionsSolve integral of ((e^x)/x)dx using basic integralsSolve integral of ((e^x)/x)dx using u-substitutionSolve integral of ((e^x)/x)dx using tabular integrationSolve integral of ((e^x)/x)dx using trigonometric substitution

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Plotting: $Ei\left(x\right)+C_0$

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a
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w
x
y
z
.
(◻)
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×
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÷
◻2
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∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Integrals of Exponential Functions

Those are integrals that involve exponential functions. Recall that an exponential function is a function of the form f(x)=a^x.

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