Rewrite the function $\cos\left(x^2\right)$ as it's representation in Maclaurin series expansion

Simplify $\left(x^2\right)^{2n}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $2n$

We can rewrite the power series as the following

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 $4n$

Multiplying fractions $\frac{{\left(-1\right)}^n}{\left(2n\right)!} \times \frac{x^{\left(4n+1\right)}}{4n+1}$

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