** Final Answer

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** Step-by-step Solution **

Problem to solve:

** Specify the solving method

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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 $T(x)$

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

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

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

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

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

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

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

Isolate $dx$ in the previous equation

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

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

Divide $1$ by $2$

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 $a \neq 1$

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

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

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

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

Isolate $dx$ in the previous equation

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

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

Divide $1$ by $2$

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

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 $a \neq 1$

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

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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

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

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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.

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

** Final Answer

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