How do you integrate #int 1/sqrt(x^2-9x-7) # using trigonometric substitution?

Question

Samantha Adkison · Accepted Answer

For this, you're going to have to complete the square. #x^2 - 9x - 7# #7 = x^2 - 9x# #28/4 + 81/4 = x^2 - 9x + 81/4# #109/4 = (x - 9/2)^2# Now bring it back into the problem. #int 1/(sqrt((x - 9/2)^2 - 109/4))dx# Let:
#u = x - 9/2#
#du = dx# #= int 1/(sqrt(u^2 - 109/4))du# Now it looks more like a #sqrt(x^2 - a^2)# form. If we let #a = sqrt(109)/2#, then: #u = sqrt(109)/2sectheta#
#du = sqrt(109)/2secthetatanthetad theta#
#sqrt(u^2 - 109/4) = sqrt(109/4sec^2theta - 109/4) = sqrt(109)/2tantheta# What we now have is: #= int 1/(cancel(sqrt(109)/2)cancel(tantheta))*cancel(sqrt(109)/2)secthetacancel(tantheta)d theta# #= int secthetad theta# Remember the trick to do this? See the following: #color(green)(intsecxdx)# #= int (secx(secx+tanx))/(secx+tanx)dx# #= int (sec^2x+secxtanx)/(secx+tanx)dx# Let:
#u = secx+tanx#
#du = secxtanx+sec^2xdx# #=> int 1/udu = ln|u| + C = color(blue)(ln|secx + tanx| + C)# Therefore, what we have is #ln|sectheta + tantheta|# for the result. Now what we can do is substitute in the proper variables. #tantheta = (2sqrt(u^2 - 109/4))/sqrt(109) = (2sqrt(x^2 - 9x - 7))/sqrt(109)#
#sectheta = (x - 9/2)*2/sqrt(109) = (2x - 9)/sqrt(109)# So the answer is: #= color(green)(ln|(2x - 9)/sqrt(109) + (2sqrt(x^2 - 9x - 7))/sqrt(109)| + C)# or, embedding the #sqrt(109)# into the #C#: #= ln|2x - 9 + 2sqrt(x^2 - 9x - 7)| - ln sqrt(109) + C# #= color(blue)(ln|2x - 9 + 2sqrt(x^2 - 9x - 7)| + C)#

Isabella Ackerman · Answer

undefined To integrate $ \int \frac{1}{\sqrt{x^2 - 9x - 7}} $ using trigonometric substitution, follow these steps:

1. Complete the square inside the square root.
   $$ x^2 - 9x - 7 = (x^2 - 9x + \frac{81}{4}) - \frac{81}{4} - 7 = (x - \frac{9}{2})^2 - \frac{121}{4} $$

2. Make the substitution $ x - \frac{9}{2} = \frac{11}{2} \sec \theta $, which implies $ x = \frac{11}{2} \sec \theta + \frac{9}{2} $, and $ dx = \frac{11}{2} \sec \theta \tan \theta \, d\theta $.

3. Substitute $ x $ and $ dx $ in terms of $ \theta $ and simplify the expression:
   $$ \frac{1}{\sqrt{x^2 - 9x - 7}} = \frac{1}{\sqrt{(\frac{11}{2} \sec \theta + \frac{9}{2})^2 - \frac{121}{4}}} $$

4. Simplify the expression inside the square root.

5. Make the trigonometric substitution and integrate the expression in terms of $ \theta $.

6. Substitute back $ x $ in terms of $ \theta $ to find the final result.