Evaluate the indefinite integral as an infinite series e^x-1/x
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Thus,
xCos(1x)=x∞∑n=0(−1)nx2n(2n)!=∞∑n=0(−1)nx2n−1(2n)!xCos(1x)=x∑n=0∞(−1)nx2n(2n)!=∑n=0∞(−1)nx2n−1(2n)!
Now integate componentwise
∫xCos(1x)dx=∫∞∑n=0(−1)nx2n−1(2n)!dx=∞∑n=0(−1)n(2n)!∫1x2n−1dx=∞∑n=0(−1)n(2n)!x2−2n2−2n+C∫xCos(1x)dx=∫∑n=0∞(−1)nx2n−1(2n)!dx=∑n=0∞(−1)n(2n)!∫1x2n−1dx=∑n=0∞(−1)n(2n)!x2−2n2−2n+C
xCos(1x)=x∞∑n=0(−1)nx2n(2n)!=∞∑n=0(−1)nx2n−1(2n)!xCos(1x)=x∑n=0∞(−1)nx2n(2n)!=∑n=0∞(−1)nx2n−1(2n)!
Now integate componentwise
∫xCos(1x)dx=∫∞∑n=0(−1)nx2n−1(2n)!dx=∞∑n=0(−1)n(2n)!∫1x2n−1dx=∞∑n=0(−1)n(2n)!x2−2n2−2n+C∫xCos(1x)dx=∫∑n=0∞(−1)nx2n−1(2n)!dx=∑n=0∞(−1)n(2n)!∫1x2n−1dx=∑n=0∞(−1)n(2n)!x2−2n2−2n+C
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