Find f if f′(x)=4/sqrt(1−x^2) and f(1/2)=5.

Answer 1

To find f, you'll need to perform the integral:

To do so, pull the constant (4) out of the integral and then realize that the derivative of arcsin(x) = 1/√(1-x²) therefore the integral above simplifies to 4arcsin(x) + c

You are given the conditions that for x = 1/2, f(x) = 5 so

5 = 4arcsin(1/2) + c

5 = 4•π/6 + c

c = 5 - 2π/3 = 15/3 - 2π/3 = (15-2π)/3

So f(x) = 4arcsin(1/2) + (15-2π)/3

Answer 2

Integrate: f(x)=4 arcsin x + C

Now use the initial condition to evaluate C.