Question

What is the value of $f(x) = \int {x{e^x}}-x dx$ if $f( - 1) = 1$ ?

Answer 1

The given function can be expressed as a product of $2$ separate functions, and this can be evaluated using Integration by parts, by splitting the function and applying the formula for integration by parts.
Formula used: Integration by parts formula - $\int {f(x)g(x)dx = f(x)\int {g(x)dx - \int {[(\dfrac{{d(f(x))}}{{dx}}} } \int {g(x)dx]} }$

Complete step-by-step solution:
The first part of the given expression, which is $x{e^x}$, can be expressed as a product of $2$ separate functions. This separation of the function is of great utility as it will be used to integrate the function. The function $x{e^x} - x$ can be split up into $2$ parts to integrate it and from there it becomes easier to break down this expression and integrate it. This approach of breaking down a complex function into individual functions and then integrating it in parts as per a given method is called Integration by parts. Now the question arises; how do we decide which functions to integrate first, and in what order do we choose functions for integration? For this process of selecting which functions to integrate first, we have a set order of function determination called ILATE, which stands for Inverse, Logarithmic, Algebraic, Trigonometric, and Exponential. This basically means that when we have a set of functions to integrate and we have to decide the order in which we pick and integrate them, we follow this pattern, that is, first we integrate the inverse function, then the logarithmic function, then the algebraic function, and so on and so forth. The formula which is used in integration by parts is,$\int {f(x)g(x)dx = f(x)\int {g(x)dx - \int {[(\dfrac{{d(f(x))}}{{dx}}} } \int {g(x)dx]} }$
Where $f(x)$ and $g(x)$ are $2$ different and distinct functions, and the order of integration is decided using the above mentioned ILATE method.
To put this formula into practice, we have
$\int {x{e^x} - xdx} = \int {x{e^x} - \int {xdx} }$
$\int {x{e^x} - xdx = \int {x{e^x}dx - \int {xdx} } }$
Now applying the integration by parts formula on the first part of the RHS of this expression we have,
$\int {x{e^x}dx = x\int {{e^x} - \int {[(\dfrac{{d(x)}}{{dx}})\int {{e^x}} dx]} } }$
On integrating the RHS we have,
$\int {} x.{e^x} = x{e^x} - \int {[1.{e^x}} dx]$
On further integration we have,
$\int {} x.{e^x} = x{e^x} - \int {} {e^x}$
The final step of integrating this first part of the function is,
$\int {} x.{e^x} = x{e^x} - {e^x}$
Hence, the first part of the given expression is now integrated. Now moving on to the full expression we have,
$\int {x{e^x} - xdx} = x{e^x} - {e^x} - \int {xdx}$
Integrating $x$ in the RHS to complete this integration by parts process, we have,
$\int {x{e^x} - xdx} = x{e^x} - {e^x} - \dfrac{{{x^2}}}{2} + c$ Hence this is the final value of the given function when fully integrated, where $c$ stands for constant.
Now, as mentioned in the question, we have to find the value of $f(x)$, if the value of $f( - 1)$ is $1$ . In order to find the complete value of$f(x)$, we have to determine the value of the constant $c$ in the expression. To find that value, we replace $x$ with $- 1$ in$f(x)$, and equate it with $1$, as mentioned in the question.
$f( - 1) = ( - 1) \times {e^{ - 1}} - {e^{ - 1}} - \dfrac{{{{( - 1)}^2}}}{2} + c$
It is given in the question that the above expression is equal to $1$. Hence equating the expression to $1$ we have,
$( - 1) \times {e^{ - 1}} - {e^{ - 1}} - \dfrac{{{{( - 1)}^2}}}{2} + c$= 1
On simplifying the above equation we have,
$- {e^{ - 1}} - {e^{ - 1}} - \dfrac{1}{2} + c = 1$
On further simplification,
$c = 1 + \dfrac{1}{2} + 2{e^{ - 1}}$
Hence, the value of constant$c$is
$c = \dfrac{3}{2} + 2{e^{ - 1}}$
Therefore, replacing $c$ with $\dfrac{3}{2} + 2{e^{ - 1}}$ in the expression, we have the complete value of $f(x)$, which is x{e^x} - {e^x} - \dfrac{{{x^2}}}{2} + $$$$\dfrac{3}{2} + 2{e^{ - 1}}.

Note: Keep in mind that after completing indefinite integration, that is integration without limits, the constant of Integration$c$must be written with the result obtained to complete the expression, otherwise it is incomplete and may lead to wrong answers.