Question

How do you find the integral of $$e^{\left(-{\displaystyle \frac{1}{2}}\right).x}$$?

Answer 1

Hint:In order to determine the integral of the above exponential function ,use the method of integration by substitution by substituting $-{\displaystyle \frac{1}{2}}(x)$ with the $u$. Find the derivative of $-{\displaystyle \frac{1}{2}}x=u$ with respect to x and put the value of $dx$ in the original integral . Use the rule of integration $\int e^{x}dx=e^c+C$ to obtain the required result.
Formula:
$\int x^{n}dx={\displaystyle \frac{x^{n+1}}{n+1}}+C$
$\int e^{x}dx=e^c+C$
${\displaystyle \frac{d}{dx}}(x)=1$

Complete step by step solution:
We are given a exponential function $$e^{\left(-{\displaystyle \frac{1}{2}}\right).x}$$ , whose integral will be
$I=\int e^{\left(-{\displaystyle \frac{1}{2}}\right).x}dx$---(1)
In order to integrate the above integral we will be using integration by substitution method by substituting $-{\displaystyle \frac{1}{2}}(x)$ with the $u$
So let $-{\displaystyle \frac{1}{2}}x=u$.
Differentiating the above with respect to $x$ using the rule of derivative that the derivative of variable $x$ is equal to one i.e. ${\displaystyle \frac{d}{dx}}(x)=1$, we get
$$\begin{gathered} {\displaystyle \frac{du}{dx}}={\displaystyle \frac{d}{dx}}\left(-{\displaystyle \frac{1}{2}}x\right) \\ {\displaystyle \frac{du}{dx}}=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dx}}\left(x\right) \\ {\displaystyle \frac{du}{dx}}=-{\displaystyle \frac{1}{2}} \\ dx=-2du \end{gathered}$$
Putting the value of $dx$ in the original integral , we get
$$\begin{gathered} I=\int e^u\left(-2du\right) \\ I=-2\int e^{u}du \end{gathered}$$
And as we know that the integral of $\int e^{x}dx=e^c+C$ where C is the constant of integration .
$$I=-2e^u+C$$
Putting back the value of $u$
$$I=\int e^{\left(-{\displaystyle \frac{1}{2}}\right).x}dx=-2e^{\left(-{\displaystyle \frac{x}{2}}\right)}+C$$
Therefore, the integral$$\int e^{\left(-{\displaystyle \frac{1}{2}}\right).x}dx$$ is equal to $$-2e^{\left(-{\displaystyle \frac{x}{2}}\right)}+C$$ where C is the constant of integration.
Additional Information:
1.Different types of methods of Integration:
Integration by Substitution
Integration by parts
Integration of rational algebraic function by using partial fraction
2. Integration by Substitution: The method of evaluating the integral by reducing it to standard form by a proper substitution is called integration by substitution.

Note:
1.Use standard formula carefully while evaluating the integrals.
2. Indefinite integral=Let $f(x)$ be a function .Then the family of all its primitives (or antiderivatives) is called the indefinite integral of $f(x)$ and is denoted by $\int f(x)dx$
3.The symbol $\int f(x)dx$ is read as the indefinite integral of $f(x)$ with respect to x.
4.Don’t forget to place the constant of integration $C$.