Final answer to the problem
Step-by-step Solution
How should I solve this problem?
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- Integrate by partial fractions
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
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Rewrite the function $\sin\left(x\right)$ 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 $\frac{\frac{{\left(-1\right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)!}}{x}$ 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 $\frac{{\left(-1\right)}^nx^{\left(2n+1\right)}}{x\left(2n+1\right)!}$ by $x$
Simplify the expression
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 $C$
