Final answer to the problem
$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^{\left(n+1\right)}x^{\left(3n+1\right)}}{\left(3n+1\right)\left(2n\right)!}+n\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^nx^{\left(3n+1\right)}}{3n\left(3n+1\right)\left(2n\right)!}+C_0$
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Step-by-step Solution
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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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1
Rewrite the function $\cos\left(x\right)$ as it's representation in Maclaurin series expansion
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$\int\left(-x^n\cos\left(x\right)+n\int x^{\left(n-1\right)}\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^n}{\left(2n\right)!}x^{2n}dx\right)dx$
Learn how to solve problems step by step online. Find the integral int(-x^ncos(x)+nint(x^(n-1)cos(x))dx)dx. Rewrite the function \cos\left(x\right) as it's representation in Maclaurin series expansion. Bring the outside term x^{\left(n-1\right)} inside the power serie. Simplify the expression. Rewrite the function \cos\left(x\right) as it's representation in Maclaurin series expansion.
Final answer to the problem
$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^{\left(n+1\right)}x^{\left(3n+1\right)}}{\left(3n+1\right)\left(2n\right)!}+n\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^nx^{\left(3n+1\right)}}{3n\left(3n+1\right)\left(2n\right)!}+C_0$
