Question

What is the antiderivative of \[{e^{ - 3x}}\] ?

Answer 1

Hint: The function is given in exponential form. We will use a substitution method. The power of the exponent will be substituted and then we will proceed to find the integration or the antiderivative as mentioned.

Complete step by step solution:
Given that,
\[{e^{ - 3x}}\]
To find the antiderivative means,
\[\int {{e^{ - 3x}}} dx\]
Now we will use substitution as ,
\[ - 3x = u\]
Taking the derivative on both sides,
\[ - 3dx = du\]
The value of dx is,
\[dx = - \dfrac{{du}}{3}\]
Now substitute in the original equation,
\[ = - \int {{e^u}\dfrac{{du}}{3}} \]
Taking the constant ratio outside,
\[ = - \dfrac{1}{3}\int {{e^u}du} \]
We know that the integration of the exponential function of this type is the function itself,
\[ = - \dfrac{1}{3}{e^u} + C\]
Now replace the value of u,
\[ = - \dfrac{1}{3}{e^{ - 3x}} + C\]
This is the correct answer.
So, the correct answer is “\[ - \dfrac{1}{3}{e^{ - 3x}} + C\]”.

Note: Note that the antiderivative is nothing but the integral. When exponential function is concerned, we know that if it is of the form \[\int {{e^x}dx} \] then the answer is definitely the function only. But if it is like if the exponent is other than this its better to use a method of substitution.
Also don’t forget to write the minus sign here in this case.
Also write the constant C at the end.