To find the antiderivatives of the given function [tex]\( f(x) = 21 e^x \)[/tex], follow these steps:
1. Identify the function to be integrated:
The given function is [tex]\( f(x) = 21 e^x \)[/tex].
2. Recall the basic integral formula for the exponential function:
The integral of [tex]\( e^x \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( e^x + C \)[/tex], where [tex]\( C \)[/tex] is a constant of integration.
3. Apply this formula to the given function:
Since the function has a constant multiplier [tex]\( 21 \)[/tex], you can factor it out of the integral:
[tex]\[
\int 21 e^x \, dx = 21 \int e^x \, dx
\][/tex]
4. Integrate the exponential function:
Using the integral of [tex]\( e^x \)[/tex], we get:
[tex]\[
\int e^x \, dx = e^x
\][/tex]
Therefore,
[tex]\[
21 \int e^x \, dx = 21 \cdot e^x
\][/tex]
5. Include the constant of integration [tex]\( C \)[/tex]:
The most general form of the antiderivative will include an arbitrary constant [tex]\( C \)[/tex]:
[tex]\[
21 e^x + C
\][/tex]
Thus, the antiderivatives of the function [tex]\( f(x) = 21 e^x \)[/tex] are:
[tex]\[
F(x) = 21 e^x + C
\][/tex]
