Question

How do you evaluate integral $$\int {\displaystyle \frac{e^{{\displaystyle \frac{1}{x}}}}{x^2}}$$

Answer 1

Hint: To solve this question, we will need some of the integrations and differentiation of functions. We should know the integration of $$\int e^{t}dt=e^t$$. Also, we should know the derivative of $${\displaystyle \frac{1}{x}}$$ with respect to x is $${\displaystyle \frac{-1}{x^2}}$$. We will use the substitution method to solve this integral.

Complete step by step solution:
Let, $${\displaystyle \frac{1}{x}}=t$$. As we know that the derivative of $${\displaystyle \frac{1}{x}}$$ with respect to x is $${\displaystyle \frac{-1}{x^2}}$$. Differentiating both sides of the expression $${\displaystyle \frac{1}{x}}=t$$, we get $${\displaystyle \frac{-1}{x^2}}dx=dt$$. Multiplying both sides by $$-1$$, we get
$$\Rightarrow {\displaystyle \frac{1}{x^2}}dx=-dt$$
We are asked to evaluate the integral $$\int {\displaystyle \frac{e^{{\displaystyle \frac{1}{x}}}}{x^2}}dx$$.
Using the above substitutions, we can replace $${\displaystyle \frac{1}{x}}=t$$ and $${\displaystyle \frac{1}{x^2}}dx=-dt$$. By doing this we get $$\int -e^{t}dt$$. So, we need to evaluate this integral now,
As $$-1$$ is a constant, it can be taken out of the integral sign. By doing this we get $$-\int e^{t}dt$$. We know that the integration $$\int e^{t}dt=e^t$$. Using this, we can evaluate the above integration as
$$\begin{array}{rl}& \Rightarrow -\int e^{t}dt=-1\times e^t+C\\ & \Rightarrow -e^t+C\end{array}$$
Here, C is the constant of integration. Replacing the t with $${\displaystyle \frac{1}{x}}$$, we get $$-e^{{\displaystyle \frac{1}{x}}}$$. Thus, the integration of $$\int {\displaystyle \frac{e^{{\displaystyle \frac{1}{x}}}}{x^2}}dx$$ is $$-e^{{\displaystyle \frac{1}{x}}}+C$$.

Note:
To solve these types of questions, one should remember the integrations and differentiation of functions. Here we used the substitution method because we could find a function and its derivative given in the expression. For indefinite integrations, it is very important to write the constant of integration in the final answer, otherwise the answer becomes incorrect.