#f(x)=intx+5sqrt(x^2+1)# #dx#
Let #x = tanu#
#dx/(du)=sec^2u#
#dx=sec^2u# #du#
#cosu=1/sqrt(1+x^2)#
#sqrt(1+x^2)=1/cosu=secu#
#intf(x)# #dx=intx+5intsecusec^2u# #du#
#intx# #dx=1/2x^2+C#
#5intsecusec^2u# #du=5(secutanu-intsecutan^2u# #du)#
#=5 secu tanu-5 int sec^3u# #du+5intsec# #du#
#10intsecusec^2u# #du=5secutanu+5ln|secu+tanu|+C#
#5intsecusec^2u# #du=1/2(5secutanu+5ln|secu+tanu|)+C#
#intf(x)# #dx=1/2(x^2+5xsqrt(x^2+1)+5ln|x+sqrt(x^2+1)|)+C#
#f(2)=7=1/2(4+10sqrt(5)+5ln|2+sqrt5|)+C#
#C=-98/10# to #2# dp.
#f(x)=1/2(x^2+5xsqrt(x^2+1)+5ln|x+sqrt(x^2+1)|)-98/10#
If you don't understand any stages, feel free to ask below.