Find out ∫ecos⁡xdx

Question

Find out  ∫ecos⁡xdx

Robert Pina · Accepted Answer

First of all: there is no close form solution in terms of elementary functions.  What can you do, but its

Anzante2m · Accepted Answer

This is just a response to your comments about having tried integration by parts for this problem.  Youre

karton · Accepted Answer

∫ecos⁡xdx=∫n=0∞cos2n⁡x(2n)!dx+∫∑n=0∞cos2n+1⁡x(2n+1)!dx=∫(1+∑n=1∞cos2n⁡x(2n)!)dx+∫∑n=0∫cos2n+1⁡x(2n+1)!dxFor n is any natural number,∫cos2n⁡xdx=(2n)!x4n(n!)2+∑k=1n(2n)!((k−1)!)2sin⁡xcos2k−1⁡x4n−k+1(n!)2(2k−1)!+CThis result can be done by successive integration by parts.For n is any non-negative integer,∫cos2n+1⁡xdx=∫cos2n⁡xd(sin⁡x)=∫(1−sin2⁡x)nd(sin⁡x)=∫∑k=0∞Ckn(−1)ksin2k⁡xd(sin⁡x)=∑k=0n(−1)kn!sin2k+1⁡xk!(n−k)!(2k+1)+C∴∫(1+∑n=1∞cos2n⁡x(2n)!)dx+∫∑n=0∞cos2n+1⁡x(2n+1)!dx=x+∑n=1∞x4n(n!)2+∑n=1∞∑k=1n(k−1)!)2sin⁡xcos2k−1⁡x4n−k+1(n!)2(2k−1)!+∑k=0n(−1)kn!sin2k+1⁡x(2n+1)!k!(n−k)!(2k+1)+C

Find out \int e^{\cos x}dx

Answered question

Answer & Explanation

New Questions in Calculus 1