Step-by-step Solution

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

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Final Answer

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

Step-by-step explanation

Problem to solve:

$\int\left(x\cdot e^{2x}\right)dx$

Choose the solving method

1

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

$u=e^{2x}$
2

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

$du=2e^{2x}dx$
3

Isolate $dx$ in the previous equation

$\frac{du}{2e^{2x}}=dx$
4

Rewriting $x$ in terms of $u$

$x=\frac{\ln\left(u\right)}{2}$
5

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

$\int\frac{\ln\left(u\right)}{4}du$
6

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

$\frac{1}{4}\int\ln\left(u\right)du$
7

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

$\frac{1}{4}\left(u\ln\left(u\right)-u\right)$
8

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

$\frac{1}{4}\left(e^{2x}\ln\left(e^{2x}\right)-e^{2x}\right)$
9

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

$\frac{1}{4}\left(2e^{2x}x-e^{2x}\right)$
10

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

$\frac{1}{4}\left(2e^{2x}x-e^{2x}\right)+C_0$
11

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

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

Final Answer

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