antiderivative.
find f.
f'(x)= 3x^(-2), f(1)=f(-1)=0
To find the antiderivative, or indefinite integral, of a function, you can use the power rule for integration. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is any real number except for -1.
In this case, the derivative of f(x) is given as f'(x) = 3x^(-2). To find the antiderivative f(x), we want to find a function whose derivative is f'(x).
Integrating the given derivative, we can use the power rule. Since the derivative is 3x^(-2), we can add 1 to the exponent, which gives -2 + 1 = -1. Then we divide by the new exponent to get -1/(-2) = 1/2. So, the antiderivative of f'(x) = 3x^(-2) is f(x) = 3*(1/2)*x^(-1), which simplifies to f(x) = (3/2)*x^(-1).
However, we still need to determine the constant of integration. We are given that f(1) = f(-1) = 0. This means that when x = 1 or x = -1, the value of f(x) is 0.
Substituting x = 1 into f(x) = (3/2)*x^(-1), we get f(1) = (3/2)*(1^(-1)) = (3/2)*(1/1) = 3/2. Since f(1) = 0, we can set the constant of integration to -3/2.
Finally, the function f(x) is:
f(x) = (3/2)*x^(-1) - 3/2