Please anyone to assist about this problem
By using hyperbolic substitution integrate sqrt (9 - x^2)
To integrate √(9 - x^2) using hyperbolic substitution, we can make use of the trigonometric identity for a hyperbolic function, which is:
cosh^2(t) - sinh^2(t) = 1
Let's start by making the substitution x = 3sinh(t), which implies dx = 3cosh(t)dt.
By substituting x and dx in terms of t, the integral becomes:
∫√(9 - (3sinh(t))^2) * 3cosh(t) dt
We can simplify the expression under the square root as follows:
√(9 - (3sinh(t))^2) = √(9 - 9sinh^2(t))
= √(9cosh^2(t))
= 3cosh(t)
The integral then simplifies to:
∫3cosh^2(t) dt
To integrate this expression, we can use the identity for the hyperbolic cosine squared:
cosh^2(t) = (cosh(2t) + 1) / 2
Substituting this into the integral, we get:
∫3[(cosh(2t) + 1)/2] dt
= (3/2)∫(cosh(2t) + 1) dt
= (3/2)∫cosh(2t) dt + (3/2)∫dt
= (3/2)sinh(2t)/2 + (3/2)t + C
Finally, we substitute back t in terms of x to obtain the final result:
= (3/2)sinh(2t)/2 + (3/2)t + C, where t = sinh^(-1)(x/3)
Therefore, the integral of √(9 - x^2) using hyperbolic substitution is given by:
(3/2)sinh[2sinh^(-1)(x/3)]/2 + (3/2)sinh^(-1)(x/3) + C