Question

Find the general indefinite integral. (Use C for the constant of integration.)


(5x + cos(x)) dx

Answer 1

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.

1. **Integral of \( 5x \)**:
\[
\int 5x \, dx = 5 \cdot \frac{x^2}{2} = \frac{5x^2}{2}
\]

2. **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:

\[
\int (5x + \cos(x)) \, dx = \frac{5x^2}{2} + \sin(x) + C
\]