Integrations Which Cannot Be Performed Analytically

Some surprisingly simple integrations cannot be found ordinarily. There is no exact expression for  
\[\int^1_0 e^{\sqrt{x^3}} dx\]
. This integration is said to be inexpressible in terms of analytic functions. The integration can be performed however by expressing  
  as a Mclaurin series and integrating term by term.
\[e^{x^3}= 1+\frac{x^3}{1!}+ \frac{x^6}{2!}+\frac{x^9}{3!}+...+ \frac{x^{3n}}{n!}+...= \sum^{\infty}_0 \frac{x^{3n}}{n!}\]
\[\int^1_0 e^{x^3} dx= \int^1_0 \sum^{\infty}_0 \frac{x^{3n}}{n!} dx = \sum^{\infty}_0[\frac{x^{3n+1}}{(3n+1)n!}]^1_0 =\sum^{\infty}_0 \frac{1}{(3n_1)n!}\]

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