Notice how #sqrt(25 + x^2) prop sqrt(a^2 + x^2)#, which implies #x = atantheta# with #a = 5#.
If we let:
#x = 5tantheta#
#dx = 5sec^2thetad theta#
#sqrt(25 + x^2) = sqrt(5^2 + 5^2tan^2theta) = 5sectheta#
Then we get:
#= int 5tantheta*5sectheta*5sec^2thetad theta#
#= 125int tanthetasecthetasec^2thetad theta#
Then, if we let:
#u = sectheta#
#du = secthetatanthetad theta#
we get:
#= 125int u^2du#
#= 125(u^3/3)#
#= 125((sec^3theta)/3)#
#= ((5^3sec^3theta)/3)#
#= ((5sectheta)^3)/3#
#= ((sqrt(25 + x^2))^3)/3#
#= color(blue)(((25 + x^2)^("3/2"))/3 + C)#