Integral of Polynomial Series

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Evaluate \[ \int_0^\infty\left(x^3-2x^5+2x^7-\frac{4}{3}x^9+\frac{2}{3}x^{11}-\frac{4}{15}x^{13}+\cdots\right)dx. \] Hint: Use Maclaurin series for exponential function.

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