Solved example of power series
Rewrite the function $\arctan\left(x\right)$ as it's representation in Maclaurin series expansion
Bring the denominator $x$ inside the power serie
Multiplying the fraction by $x^{\left(2n+1\right)}$
Divide fractions $\frac{\frac{{\left(-1\right)}^nx^{\left(2n+1\right)}}{2n+1}}{x}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$
Simplify the fraction $\frac{{\left(-1\right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)x}$ by $x$
Simplify the expression inside the integral
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
Multiplying the fraction by ${\left(-1\right)}^n$
Any expression multiplied by $1$ is equal to itself
Simplify the expression inside the integral
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 $\frac{{\left(-1\right)}^n}{2n+1} \times \frac{x^{\left(2n+1\right)}}{2n+1}$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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