ðŸ‘‰ Try now NerdPal! Our new math app on iOS and Android

# Find the integral $\int xe^{2x}dx$

## Step-by-step Solution

Go!
Math mode
Text mode
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
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

###  Videos

$\frac{1}{2}xe^{2x}-\frac{1}{4}e^{2x}+C_0$
Got another answer? Verify it here!

##  Step-by-step Solution 

Problem to solve:

$\int xe^{2x}dx$

Specify the solving method

1

We can solve the integral $\int xe^{2x}dx$ by applying the method of tabular integration by parts, which allows us to perform successive integrations by parts on integrals of the form $\int P(x)T(x) dx$. $P(x)$ is typically a polynomial function and $T(x)$ is a transcendent function such as $\sin(x)$, $\cos(x)$ and $e^x$. The first step is to choose functions $P(x)$ and $T(x)$

$\begin{matrix}P(x)=x \\ T(x)=e^{2x}\end{matrix}$

Find the derivative of $x$ with respect to $x$

$x$

The derivative of the linear function is equal to $1$

$1$

The derivative of the constant function ($1$) is equal to zero

0
2

Derive $P(x)$ until it becomes $0$

$0$

Find the integral of $e^{2x}$ with respect to $x$

$e^{2x}$

We can solve the integral $\int e^{2x}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=2x$

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

$du=2dx$

Isolate $dx$ in the previous equation

$du=2dx$

Substituting $u$ and $dx$ in the integral and simplify

$\int\frac{e^u}{2}du$

Take the constant $\frac{1}{2}$ out of the integral

$\frac{1}{2}\int e^udu$

Divide $1$ by $2$

$\frac{1}{2}\int e^udu$

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 $a \neq 1$

$\frac{1}{2}e^u$

Replace $u$ with the value that we assigned to it in the beginning: $2x$

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

The integral of a function times a constant ($\frac{1}{2}$) is equal to the constant times the integral of the function

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

We can solve the integral $\int e^{2x}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=2x$

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

$du=2dx$

Isolate $dx$ in the previous equation

$du=2dx$

Substituting $u$ and $dx$ in the integral and simplify

$\frac{1}{2}\int\frac{e^u}{2}du$

Take the constant $\frac{1}{2}$ out of the integral

$\frac{1}{2}\cdot \left(\frac{1}{2}\right)\int e^udu$

Divide $1$ by $2$

$\frac{1}{2}\cdot \frac{1}{2}\int e^udu$

Multiply $\frac{1}{2}$ times $\frac{1}{2}$

$\frac{1}{4}\int e^udu$

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 $a \neq 1$

$\frac{1}{4}e^u$

Replace $u$ with the value that we assigned to it in the beginning: $2x$

$\frac{1}{4}e^{2x}$
3

Integrate $T(x)$ as many times as we have had to derive $P(x)$, so we must integrate $e^{2x}$ a total of $2$ times

$\frac{1}{4}e^{2x}$
4

With the derivatives and integrals of both functions we build the following table

$\begin{matrix}\mathrm{Derivatives} & \mathrm{Sign} & \mathrm{Integrals} \\ & & e^{2x} \\ x & + & \frac{1}{2}e^{2x} \\ 1 & - & \frac{1}{4}e^{2x} \\ 0 & & \end{matrix}$
5

Then the solution is the sum of the products of the derivatives and the integrals according to the previous table. The first term consists of the product of the polynomial function by the first integral. The second term is the product of the first derivative by the second integral, and so on.

$\frac{1}{2}xe^{2x}-\frac{1}{4}e^{2x}$
6

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

$\frac{1}{2}xe^{2x}-\frac{1}{4}e^{2x}+C_0$

$\frac{1}{2}xe^{2x}-\frac{1}{4}e^{2x}+C_0$

##  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 xe^2xdx using partial fractionsSolve integral of xe^2xdx using basic integralsSolve integral of xe^2xdx using u-substitutionSolve integral of xe^2xdx using integration by partsSolve integral of xe^2xdx using trigonometric substitution

SnapXam A2

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
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

### Main topic:

Integrals of Exponential Functions

###  Join 500k+ students in problem solving.

##### Without automatic renewal.
Create an Account