## Final Answer

## Step-by-step explanation

Problem to solve:

Choose the solving method

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 $e^{2x}$ 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

Rewriting $x$ in terms of $u$

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

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

The integral of the natural logarithm is given by the following formula, $\displaystyle\int\ln(x)dx=x\ln(x)-x$

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

Apply the formula: $\ln\left(e^x\right)$$=x$, where $x=2x$

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

Solve the product $\frac{1}{4}\left(2e^{2x}x-e^{2x}\right)$