#sqrt(x^2 - 49) prop sqrt(x^2 - a^2)#
Let:
#x = asectheta# with #a = 7#:
#dx = 7secthetatanthetad theta#
#sqrt(x^2 - 49) = sqrt(7^2sec^2theta - 7^2) = 7sqrt(sec^2theta - 1)#
#= 7tantheta#
Thus:
#= int 7tantheta * 7secthetatanthetad theta#
#= 49int secthetatan^2thetad theta#
#= 49int sectheta(sec^2theta - 1)d theta#
#= color(highlight)(49)int sec^3theta- secthetad theta#
Solving these individually:
Small trick:
#int secthetad theta = int sectheta((sectheta + tantheta)/(sectheta + tantheta))d theta#
#= int (sec^2theta + secthetatantheta)/(sectheta + tantheta)d theta#
u-substitution. Let:
#u = sectheta + tantheta#
#du = secthetatantheta + sec^2thetad theta#
#=> int 1/udu#
#= ln|u| = color(green)(ln|sectheta + tantheta|)#
The other one:
#int sec^3thetad theta#
Integration by Parts. Let:
#u = sectheta#
#du = secthetatantheta#
#dv = sec^2thetad theta#
#v = tantheta#
#=> uv - int vdu#
#= secthetatantheta - int secthetatan^2thetad theta#
#= secthetatantheta - int sec^3theta - secthetad theta#
#int sec^3thetad theta = secthetatantheta - int sec^3thetad theta + int secthetad theta#
Add the #sec^3theta# over. No need to evaluate it. We also know #intsecthetad theta# already.
#2int sec^3thetad theta = secthetatantheta + int secthetad theta#
#2int sec^3thetad theta = secthetatantheta + ln|sectheta + tantheta|#
#int sec^3thetad theta = color(green)(1/2[secthetatantheta + ln|sectheta + tantheta|])#
Overall:
#= overbrace(1/2[secthetatantheta + ln|sectheta + tantheta|])^(intsec^3thetad theta) - overbrace(ln|sectheta + tantheta|)^(intsecthetad theta)#
#= 1/2[secthetatantheta - ln|sectheta + tantheta|]#
Recall that #sectheta = x/7# and #tantheta = sqrt(x^2 - 49)/7#, and don't forget the #color(highlight)(49)#:
#= 49{1/2[x/7*sqrt(x^2 - 49)/7 - ln|x/7 + sqrt(x^2 - 49)/7|]}#
#= 1/2[(xsqrt(x^2 - 49)) - 49ln|x/7 + sqrt(x^2 - 49)/7|]#
#= 1/2[(xsqrt(x^2 - 49)) - 49ln|(1/7)(x + sqrt(x^2 - 49))|]#
#= 1/2[(xsqrt(x^2 - 49)) - 49(ln|x + sqrt(x^2 - 49)| + ln(1/7))] + C#
The remaining #-49/2ln(1/7)# gets embedded into #C#:
#= color(blue)(1/2[xsqrt(x^2 - 49) - 49ln|x + sqrt(x^2 - 49)|] + C)#
