Rewrite the function $\sin\left(x\right)$ as it's representation in Maclaurin series expansion

Bring the denominator $x$ inside the power serie

We can rewrite the power series as the following

The integral of a function times a constant (${\left(-1\right)}^n$) is equal to the constant times the integral of the function

Multiply the fraction by the term

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2n$

Multiplying fractions $\frac{{\left(-1\right)}^n}{\left(2n+1\right)!} \times \frac{x^{\left(2n+1\right)}}{2n+1}$

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$