Here, we show you a step-by-step solved example of calculus. This solution was automatically generated by our smart calculator:
We can solve the integral $\int xe^{2x}dx$ 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 or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
We can solve the integral $\int e^{2x}dx$ 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
Rearrange the equation
Divide both sides of the equation by $2$
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of 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 $a \neq 1$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
The integral of a function times a constant ($\frac{1}{2}$) is equal to the constant times the integral of the function
Multiplying the fraction by $-1$
Now replace the values of $u$, $du$ and $v$ in the last formula
We can solve the integral $\int e^{2x}dx$ 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
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
Multiplying fractions $-\frac{1}{2} \times \frac{1}{2}$
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 $-\frac{1}{2}\int e^{2x}dx$ 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 $C$
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