Discover how IDNLearn.com can help you find the answers you need quickly and easily. Discover trustworthy solutions to your questions quickly and accurately with help from our dedicated community of experts.
Sagot :
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.
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.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.
