Final Answer
$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(2n+1\right)!}+C_0$
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Step-by-step Solution
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1
Rewrite the function $\sin\left(x\right)$ as it's representation in Maclaurin series expansion
$\int\frac{\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^n}{\left(2n+1\right)!}x^{\left(2n+1\right)}}{x}dx$
2
Bring the denominator $x$ inside the power serie
$\int\sum_{n=0}^{\infty } \frac{\frac{{\left(-1\right)}^n}{\left(2n+1\right)!}x^{\left(2n+1\right)}}{x}dx$
Intermediate steps
3
Simplify the expression inside the integral
$\int\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^nx^{2n}}{\left(2n+1\right)!}dx$
Explain this step further
4
We can rewrite the power series as the following
$\sum_{n=0}^{\infty } \frac{1}{\left(2n+1\right)!}\int{\left(-1\right)}^nx^{2n}dx$
5
The integral of a function times a constant (${\left(-1\right)}^n$) is equal to the constant times the integral of the function
$\sum_{n=0}^{\infty } \frac{1}{\left(2n+1\right)!}{\left(-1\right)}^n\int x^{2n}dx$
Intermediate steps
6
Simplify the expression inside the integral
$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^n}{\left(2n+1\right)!}\int x^{2n}dx$
Explain this step further
7
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 $2n$
$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^n}{\left(2n+1\right)!}\frac{x^{\left(2n+1\right)}}{2n+1}$
8
Multiplying fractions $\frac{{\left(-1\right)}^n}{\left(2n+1\right)!} \times \frac{x^{\left(2n+1\right)}}{2n+1}$
$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(2n+1\right)!}$
9
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(2n+1\right)!}+C_0$
Final Answer
$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(2n+1\right)!}+C_0$