Final Answer
Step-by-step Solution
Problem to solve:
Specify 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 $2x$ 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 $1$
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
Divide both sides of the equation by $2$
Simplifying the quotients
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: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
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
The derivative of the linear function is equal to $1$
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
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 $\left(e^u\cdot u-\int e^udu\right)$
Multiply $2$ times $\frac{1}{4}$
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: $2x$
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 $C$