** 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 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

Differentiate both sides of the equation $u=2x$

Find the derivative

The derivative of the linear function times a constant, is equal to the constant

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

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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

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Isolate $dx$ in the previous equation

Divide both sides of the equation by $2$

Simplifying the quotients

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Rewriting $x$ in terms of $u$

Rewriting $x$ in terms of $\frac{u}{2}$

Multiplying the fraction by $e^u$

Divide fractions $\frac{\frac{ue^u}{2}}{2}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

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

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Take the constant $\frac{1}{4}$ out of the integral

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Divide $1$ by $4$

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We can solve the integral $\int ue^udu$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

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

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First, identify $u$ and calculate $du$

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Now, identify $dv$ and calculate $v$

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Solve the integral

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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 $a \neq 1$

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

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Multiply the single term $\frac{1}{4}$ by each term of the polynomial $\left(e^u\cdot u-\int e^udu\right)$

Multiply $2$ times $\frac{1}{4}$

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

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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The integral $-\frac{1}{4}\int e^udu$ results in: $-\frac{1}{4}e^{2x}$

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Gather the results of all integrals

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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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