Final answer to the problem
$\sum_{n=0}^{\infty } \frac{{\left(-\pi \right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(n!\right)}+C_0$
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Step-by-step Solution
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- Integrate by partial fractions
- Integrate by substitution
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- Integrate using tabular integration
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- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
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1
Rewrite the function $e^{-\pi x^2}$ as it's representation in Maclaurin series expansion
$\int\sum_{n=0}^{\infty } \frac{\left(-\pi x^2\right)^n}{n!}dx$
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$\int\sum_{n=0}^{\infty } \frac{\left(-\pi x^2\right)^n}{n!}dx$
Learn how to solve problems step by step online. Find the integral int(e^(-pix^2))dx. Rewrite the function e^{-\pi x^2} as it's representation in Maclaurin series expansion. The power of a product is equal to the product of it's factors raised to the same power. Simplify \left(x^2\right)^n 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 n. We can rewrite the power series as the following.
Final answer to the problem
$\sum_{n=0}^{\infty } \frac{{\left(-\pi \right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(n!\right)}+C_0$
