Question

Find the integral: $\int {\dfrac{{{e^x}}}{x}\left( {1 + x\ln x} \right)dx}$.

Answer 1

Divide by x in the numerator and denominator and then assume $f\left( x \right) = \ln x$. The integral will transform into $\int {{e^x}\left[ {f\left( x \right) + f'\left( x \right)} \right]dx}$ form and the integration of this form is $\int {{e^x}\left[ {f\left( x \right) + f'\left( x \right)} \right]dx} = {e^x}f\left( x \right) + c$ which will be the desired result.

Complete step by step answer:
Let us assume the given integral as
$\Rightarrow I = \int {\dfrac{{{e^x}}}{x}\left( {1 + x\ln x} \right)dx}$
Now, divide the numerator and denominator by $x$, we get,
$\Rightarrow I = \int {{e^x}\left( {\dfrac{1}{x} + \ln x} \right)dx}$
Now, if we consider, $f\left( x \right) = \ln x$, then,
$\Rightarrow f'\left( x \right) = \dfrac{d}{{dx}}\left( {\ln x} \right)$
On differentiating the term on the right side, we get
$\Rightarrow f'\left( x \right) = \dfrac{1}{x}$
So, we can write the above integral I as,
$\Rightarrow I = \int {{e^x}\left[ {f\left( x \right) + f'\left( x \right)} \right]dx}$
Now, we know that,
$\int {{e^x}\left[ {f\left( x \right) + f'\left( x \right)} \right]dx} = {e^x}f\left( x \right) + c$
Using this property, the above integral value will be,
$\Rightarrow I = {e^x}f\left( x \right) + c$
Put back the value $f\left( x \right) = \ln x$ in the above integral I, we get
$\therefore I = {e^x}\ln x + c$

Hence the integration of $\int {\dfrac{{{e^x}}}{x}\left( {1 + x\ln x} \right)dx}$ is ${e^x}\ln x + c$.

Additional Information: Differentiation and integration are the two important concepts of calculus. Calculus is a branch of mathematics that deals with the study of problems involving a continuous change in the values of quantities. Differentiation refers to simplifying a complex function into simpler functions. Integration generally refers to summing up the smaller function to form a bigger unit.
Indefinite integrals are those integrals that do not have any limit of integration. It has an arbitrary constant. Definite integrals are those integrals which have an upper and lower limit. Definite integral has two different values for the upper limit and lower limit when they are evaluated. The final value of a definite integral is the value of integral to the upper limit minus the value of the definite integral for the lower limit.

Note: To solve these types of questions one should know the basic concepts of integral calculus. Also, it is important to note how we have substituted the value of $f\left( x \right) = \ln x$ by realizing the fact that the integral given to us is a summation of the function and differentiation of this and hence if we substitute this, the problem will be simplified to a great extent. Students should remember the differentiation formulas i.e. $\dfrac{d}{{dx}}\left( {\ln x} \right) = \dfrac{1}{x}$.