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Here, we show you a step-by-step solved example of power series. This solution was automatically generated by our smart calculator:

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

Simplify $\left(x^2\right)^{\left(2n+1\right)}$ 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 $2n+1$

Solve the product $2\left(2n+1\right)$

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 $4n+2$

Multiplying fractions $\frac{{\left(-1\right)}^n}{\left(2n+1\right)!} \times \frac{x^{\left(4n+3\right)}}{4n+3}$

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