Question

Find the most general antiderivative of f(x) = -9x - 8(1 - x^2)^{-1/2} , where -1 < x < 1 .

a) -9x + 8√1 - x^2 + C
b) -9x - 8√1 - x^2 + C
c) -9x + 8 arcsin(x) + C
d) -9x - 8 arcsin(x) + C

Answer 1

The most general antiderivative of the function f(x) = -9x - 8(1 - x^2)^{-1/2} is -9x - 8 arcsin(x) + C, where C is the constant of integration.

Step-by-step explanation:

To find the most general antiderivative (indefinite integral) of the function f(x) = -9x - 8(1 - x2)-1/2, where -1 < x < 1, we tackle each term separately.

The antiderivative of -9x is straightforward:
- (9/2)x2 + C1

Now for the second term, -8(1 - x^2)^-1/2, we recognize this as the derivative of the arcsin function. The antiderivative of (1 - x2)-1/2 is arcsin(x), so when we multiply by -8, we get:
- 8 arcsin(x) + C2

Combining both terms, we get the general antiderivative:

-9x - 8 arcsin(x) + C

Since C1 and C2 are constants of integration, we can combine them into a single constant C.