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)
