To find all the antiderivatives of the function [tex]\( f(x) = 9 \sin x - 2 \)[/tex], we need to integrate the function. Let's break down the process step-by-step:
1. Identify the components of the function:
[tex]\( f(x) = 9 \sin x - 2 \)[/tex]
2. Integrate each term separately:
- The integral of [tex]\( 9 \sin x \)[/tex]:
[tex]\[
\int 9 \sin x \, dx
\][/tex]
To integrate [tex]\( 9 \sin x \)[/tex], we know from calculus that the integral of [tex]\( \sin x \)[/tex] is [tex]\( -\cos x \)[/tex]. Therefore,
[tex]\[
\int 9 \sin x \, dx = 9 \left( \int \sin x \, dx \right) = 9 \left( -\cos x \right) = -9 \cos x
\][/tex]
- The integral of [tex]\(-2\)[/tex]:
[tex]\[
\int -2 \, dx
\][/tex]
The integral of a constant [tex]\( k \)[/tex] is [tex]\( kx \)[/tex]. Therefore,
[tex]\[
\int -2 \, dx = -2x
\][/tex]
3. Combine the integrals:
Combining these results, we get the general antiderivative:
[tex]\[
F(x) = -9 \cos x - 2x + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
4. Check the work by taking the derivative of [tex]\( F(x) \)[/tex]:
[tex]\[
F(x) = -9 \cos x - 2x + C
\][/tex]
To check, compute:
[tex]\[
F'(x) = \frac{d}{dx} \left( -9 \cos x - 2x + C \right)
\][/tex]
- The derivative of [tex]\( -9 \cos x \)[/tex] is [tex]\( 9 \sin x \)[/tex].
- The derivative of [tex]\( -2x \)[/tex] is [tex]\( -2 \)[/tex].
- The derivative of the constant [tex]\( C \)[/tex] is [tex]\( 0 \)[/tex].
Therefore,
[tex]\[
F'(x) = 9 \sin x - 2
\][/tex]
This derivative matches the original function [tex]\( f(x) \)[/tex].
Thus, the antiderivative of [tex]\( f(x) = 9 \sin x - 2 \)[/tex] is:
[tex]\[
F(x) = -9 \cos x - 2x + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
