** Final answer to the problem

**

** Step-by-step Solution **

** How should I solve this problem?

- Integrate by substitution
- Integrate by partial fractions
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- FOIL Method
- Load more...

**

**

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

**

**

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

**

**

Divide $1$ by $4$

**

**

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

**

**

First, identify or choose $u$ and calculate it's derivative, $du$

**

**

Now, identify $dv$ and calculate $v$

**

**

Solve the integral to find $v$

**

**

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 $a \neq 1$

**

**

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

**

**

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

**

**

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

**

**

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

**

**

Gather the results of all integrals

**

**

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 to the problem

**