To find the indefinite integral of \( (5x + \cos(x)) , dx \), we can break it down into two separate integrals:
\[ \int (5x + \cos(x)) , dx = \int 5x , dx + \int \cos(x) , dx \]
Now, we will compute each integral separately.
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Integral of \( 5x \): \[ \int 5x , dx = 5 \cdot \frac{x^2}{2} = \frac{5x^2}{2} \]
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Integral of \( \cos(x) \): \[ \int \cos(x) , dx = \sin(x) \]
Now, we can combine the results:
\[ \int (5x + \cos(x)) , dx = \frac{5x^2}{2} + \sin(x) + C \]
where \( C \) is the constant of integration.
Thus, the final result is:
\[ \int (5x + \cos(x)) , dx = \frac{5x^2}{2} + \sin(x) + C \]