Intergral (27-9x^2)^(1/2)
fit to theorem (a^2-u^2)^(1/2)
To integrate the expression (27-9x^2)^(1/2), we can utilize the theorem for integrating the form (a^2-u^2)^(1/2).
According to this theorem, the integral of (a^2-u^2)^(1/2) with respect to u is given by arcsin(u/a) + C, where C is the constant of integration.
Now, let's compare the given expression (27-9x^2)^(1/2) with the form (a^2-u^2)^(1/2).
In the given expression, a^2 is 27 and u^2 is 9x^2. To match the form (a^2-u^2)^(1/2), we need to factor out a 9 from the expression under the square root:
(27-9x^2)^(1/2) can be rewritten as (9(3-x^2))^(1/2).
Now, we can rewrite the expression as 3(1 - (x^2/3))^(1/2), where a^2 = 3 and u^2 = x^2/3.
Comparing this new expression to the form (a^2-u^2)^(1/2), we see that a^2 is 3 and u^2 is x^2/3.
According to the theorem, the integral of (27-9x^2)^(1/2) is arcsin(x/√3) + C, where C is the constant of integration.
Hence, the integral of (27-9x^2)^(1/2) is arcsin(x/√3) + C.