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Question: How do you find the antiderivative of \({e^{ - x}}\)?...

How do you find the antiderivative of e−x{e^{ - x}}?

Explanation

Solution

Given an expression. We have to find the antiderivative of the expression. The antiderivative means the integration of the expression. We will first assume the variable as some temporary variable. Then differentiate the expression. Then, substitute the values into the integral and apply the exponential rule of integration.

Formula used:
The integral of exponential function is given by:
∫exdx=ex+C\int {{e^x}dx} = {e^x} + C

Complete step by step solution:
We are given the expression e−x{e^{ - x}}. We have to find the antiderivative of the function.
⇒∫e−xdx\Rightarrow \int {{e^{ - x}}dx}
Substitute −x=u - x = u into the expression.
Differentiate both sides of equation −x=u - x = u with respect to x.
⇒−ddxx=ddxu\Rightarrow - \dfrac{d}{{dx}}x = \dfrac{d}{{dx}}u
⇒−1=dudx\Rightarrow - 1 = \dfrac{{du}}{{dx}}
⇒dx=−du\Rightarrow dx = - du
Now, we will substitute the value of x and dx into the integral function.
⇒∫−eudu\Rightarrow \int { - {e^u}du}
Take out negative sign from the integral, we get:
⇒−∫eudu\Rightarrow - \int {{e^u}du}
Now, integrate the exponential function, we get:
⇒−eu+C\Rightarrow - {e^u} + C
Now, replace uu by −x - x into the expression.
⇒−e−x+C\Rightarrow - {e^{ - x}} + C

Final answer: Hence, the antiderivative of the expression is equal to −e−x+C - {e^{ - x}} + C

Additional information:
Antiderivative of the particular function is equivalent to the integration of the function. This process is opposite to the method of differentiation. The antiderivative of every continuous function always exists. There are many functions that have more than one antiderivative. These antiderivatives differ from each other by a constant value. The derivative and antiderivative of the exponential function are always equal to the function itself.

Note:
In such types of questions the students mainly don't get an approach on how to solve it. In such types of questions students mainly forget to first substitute the exponent to another variable. Also, students can make mistakes while differentiating the variable with respect to another variable.