Certainly! Let's solve the integral of the function [tex]\( 5 \cos(x) + 4 \sin(x) \)[/tex] step by step. We'll integrate with respect to [tex]\( x \)[/tex].
The integral we need to solve is:
[tex]\[
\int (5 \cos(x) + 4 \sin(x)) \, dx
\][/tex]
To perform the integration, we can split the integral into two separate integrals:
[tex]\[
\int (5 \cos(x) + 4 \sin(x)) \, dx = \int 5 \cos(x) \, dx + \int 4 \sin(x) \, dx
\][/tex]
Now, let’s integrate each part individually.
### Integrating [tex]\( 5 \cos(x) \)[/tex]:
The integral of cosine is a standard integral:
[tex]\[
\int \cos(x) \, dx = \sin(x)
\][/tex]
Therefore:
[tex]\[
\int 5 \cos(x) \, dx = 5 \int \cos(x) \, dx = 5 \sin(x)
\][/tex]
### Integrating [tex]\( 4 \sin(x) \)[/tex]:
The integral of sine is another standard integral:
[tex]\[
\int \sin(x) \, dx = -\cos(x)
\][/tex]
Therefore:
[tex]\[
\int 4 \sin(x) \, dx = 4 \int \sin(x) \, dx = 4 (- \cos(x)) = -4 \cos(x)
\][/tex]
### Combining the results:
Now we combine the results of the two integrals:
[tex]\[
\int (5 \cos(x) + 4 \sin(x)) \, dx = 5 \sin(x) - 4 \cos(x)
\][/tex]
Finally, we add the constant of integration [tex]\( C \)[/tex] to account for the indefinite integral:
[tex]\[
\int (5 \cos(x) + 4 \sin(x)) \, dx = 5 \sin(x) - 4 \cos(x) + C
\][/tex]
So, the integral of [tex]\( 5 \cos(x) + 4 \sin(x) \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
5 \sin(x) - 4 \cos(x) + C
\][/tex]
