Answer:
The value of the integral ∫ dx / (x√x^2 - 36) is -x/6 + C.
Explanation:
To evaluate the integral ∫ dx / (x√x^2 - 36) using the substitution
x = 6sect, we can follow these steps:
Step 1:
Find the derivative of x = 6sect with respect to t.
dx = 6sec(t)tan(t) dt
Step 2:
Substitute the expression for dx and x into the integral, and simplify.
∫ (6sec(t)tan(t) dt) / (6sect * √(36sec^2(t) - 36))
Simplifying further:
∫ sec(t)tan(t) dt / (√(36sec^2(t) - 36))
∫ tan(t) dt / (√(sec^2(t)))
Step 3:
Simplify using trigonometric identities.
∫ tan(t) dt / (sec(t))
∫ sin(t) / cos(t) dt
Step 4:
Apply the substitution u = cos(t), du = -sin(t) dt.
∫ -du
-u + C
Step 5:
Substitute back u = cos(t).
-cos(t) + C
Step 6:
Substitute back x = 6sect.
-cos(t) + C
-cos(arccos(x/6)) + C
-x/6 + C
Therefore,
The integral ∫ dx / (x√x^2 - 36) evaluated using the given substitution
x = 6sect is -x/6 + C.