Question

What is the integral of $2x{e^x}?$

Answer 1

Hint : As we can see that we have to solve the given integral. We can solve this integral by using the formula of integration by parts and doing some calculations we will get the required answer. We will be using the formula which is $\int {udv = uv - \int {vdu} }$ .

Complete step-by-step answer :
Here we have to find the integral of $2x{e^x}$ .
So we can write it as $\int {2x{e^x}dx}$ . We have to solve this.
We can see that there is a constant value in the expression i.e. $2$ . Since it is an integral value we can take it outside the integral sign and thus the equation can be re-written as $2\int {x{e^x}dx}$ .
Let us assume that $I = \int {x{e^x}dx}$ . So we can write $2\int {x{e^x}dx}$ as $2I$ .
Now by applying the formula of integral by parts and by comparing we have $u \to x$ , so we say $du = dx$ .
And we have $dv = {e^x}dx$ , therefore $v = {e^x}$ .
By substituting the values in the formula we can write
$\int {x{e^x}dx = x{e^x} - \int {{e^x}dx} }$ .
We can cancel out the common terms, thus it gives $\int {{e^x}dx = {e^x}}$ . On further integrating we can write $\int {x{e^x}dx = x{e^x} - {e^x}}$ .
From the above we have $I = \int {x{e^x}dx}$ , so we can rewrite it as $I = x{e^x} - {e^x}$ .
Therefore we have $2\int {x{e^x}dx}$ as $2I$ , so $2\int {x{e^x}dx}$ $= 2(x{e^x} - {e^x}) + C$ .
Hence the required integral value of $2\int {x{e^x}dx}$ is $2(x{e^x} - {e^x}) + C$ .
So, the correct answer is “ $2(x{e^x} - {e^x}) + C$ ”.

Note : We should note that the integral of ${e^x}$ with respect to $x$ is ${e^x}$ . Before solving this kind of question we should be fully aware of the integration and their formulas. We should avoid the calculation mistake. All the basic integration and derivative formulas should be memorized to solve these types of questions.