Question

Find the value of $\int {x{e^x}dx} $.

Answer 1

Hint: In this question, we are given an expression and we have to integrate that expression. Use integration by parts to solve the question. At first, identify the first and second parts of the question using ILATE. Then, apply the formula. Keep the first part aside and integrate the second part. Subtract from it the integration of the product of differentiation of the first part and integration of second. Use this formula to find the answer. Do not forget to add the constant of integration at the end.

Formula used: $\int {uvdx = u\int {vdx - \int {\dfrac{{du}}{{dx}}} } } \left( {\int {vdx} } \right)dx$

Complete step-by-step solution:
We have to integrate the given expression. We can see that there are two terms which have been multiplied with each other. In this case, we use Integration by parts.
The formula for integration by parts is –
$\int {uvdx = u\int {vdx - \int {\dfrac{{du}}{{dx}}} } } \left( {\int {vdx} } \right)dx$
In this method, we choose the values of $u$ and $v$ by ILATE method. ILATE stands for –
I – Inverse trigonometric function
L – Logarithmic function
A – Algebraic function
T – Trigonometric function
E – Exponential function
The function appearing first in ILATE is chosen to be $u$ and the function appearing second is chosen to be $v$.
In the given expression, we have an algebraic (A) and an exponential (E) function. Since A comes before E, we will consider $u = x$ and $v = {e^x}$.
Using the integration by parts formula to integrate –
$ \Rightarrow \int {x{e^x}dx} $
Let $u = x$ and $v = {e^x}$. Using by parts formula,
$ \Rightarrow x\int {{e^x}dx - \int {\dfrac{{d\left( x \right)}}{{dx}}\left( {\int {{e^x}dx} } \right)} } dx$
Simplifying the equation –
$ \Rightarrow x{e^x} - \int {{e^x}dx} $
Hence,
$ \Rightarrow x{e^x} - {e^x} + C$

$\therefore $ The value of $\int {x{e^x}dx} = x{e^x} - {e^x} + C$

Note: This method of integration is generally used when two functions are multiplied together. It must be observed that we do not add the constant of integration while integrating ${e^x}$ both the times. This is because it is added only at the end so that there is no repetition. This shows that adding a constant to the integral of the second function is superfluous so far as the final result is concerned while applying the method of integration by parts.
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.