Expression 3: StartFraction, "d" Over "d" "x" , EndFraction natural log left parenthesis, sinh left parenthesis, "x" , right parenthesis , right parenthesis equals StartFraction, 1 Over "x" , EndFraction plus Start sum from "n" equals 1 to backslash "i" "n" "f" "t" "y" , end sum, StartFraction, 1 Over 1 plus StartNestedFraction, "x" squared NestedOver "n" squared pi squared , EndNestedFraction , EndFraction times StartFraction, 2 "x" Over "n" squared pi squared , EndFraction equals StartFraction, 1 Over "x" , EndFraction plus 2 "x" Start sum from "n" equals 1 to backslash "i" "n" "f" "t" "y" , end sum, StartFraction, 1 Over "n" squared pi squared plus "x" squared , EndFractionddxlnsinhx=1x+\infty∑n=111+x2n2π2·2xn2π2=1x+2x\infty∑n=11n2π2+x2
3
Expression 4: 1 plus StartFraction, 2 Over "e" Superscript, 2 "x" , Baseline minus 1 , EndFraction equals StartFraction, 1 Over "x" , EndFraction plus 2 "x" Start sum from "n" equals 1 to backslash "i" "n" "f" "t" "y" , end sum, StartFraction, 1 Over "n" squared pi squared plus "x" squared , EndFraction1+2e2x−1=1x+2x\infty∑n=11n2π2+x2
4
Expression 5: Start sum from "n" equals 1 to backslash "i" "n" "f" "t" "y" , end sum, StartFraction, 1 Over "n" squared pi squared plus "x" squared , EndFraction equals StartFraction, 1 Over 2 "x" , EndFraction left parenthesis, 1 plus StartFraction, 2 Over "e" Superscript, 2 "x" , Baseline minus 1 , EndFraction minus StartFraction, 1 Over "x" , EndFraction , right parenthesis\infty∑n=11n2π2+x2=12x1+2e2x−1−1x
