How do you integrate the function 1 + sin(x)?

To integrate the function 1 + sin(x), we will proceed step by step.

The integral we need to solve is:

∫ (1 + sin(x)) dx

This integral can be split into two separate integrals:

∫ 1 dx + ∫ sin(x) dx

Now, let’s solve each part:

1. Integrating 1:

The integral of 1 with respect to x is simply:

∫ 1 dx = x

2. Integrating sin(x):

The integral of sin(x) is:

∫ sin(x) dx = -cos(x)

Putting it all together:

Now, we can combine the results:

∫ (1 + sin(x)) dx = x - cos(x) + C

Here, C represents the constant of integration, which is included because the integral can have infinitely many solutions that differ by a constant.

Thus, the final answer is:

∫ (1 + sin(x)) dx = x - cos(x) + C

This result shows the antiderivative of the function 1 + sin(x), providing both a clear methodology for integration and the final solution.