#Does anyone know how to solve this?
viscid merlin
spiral venture
you should've write sqrt(16-x²)/16x + c
cuz that's your final answer
also the variable there is different should be (1/16)cot(ø) not (1/16)cot(x)
and the variable in the question is x not ø, so you should write in terms of x
obtuse night
d/dx sin(x) = cos(x), not -cos(x).
After that, then yeah, the problem is that that's not the integral in terms of x.
worldly geyser
Here's a way without trigonometric substitution. And a bit more general.
dx/(x^2 √(a^2 - x^2)) = dx/(x^2 |x|√(a^2/x^2 - 1)) = sgn(x)dx/(x^3√(a^2/x^2 - 1)) = -(1/(2a^2))sgn(x)d(a^2/x^2)/√(a^2/x^2 - 1)
Now, let's look at two cases.
Case 1. x > 0.
-(1/(2a^2))d(a^2/x^2)/√(a^2/x^2 - 1)
The antiderivative is -(1/a^2)√(a^2/x^2 - 1) = -(1/a^2)√(a^2 - x^2)/x.
Case 2. x < 0.
(1/(2a^2))d(a^2/x^2)/√(a^2/x^2 - 1)
The antiderivative is (1/a^2)√(a^2/x^2 - 1) = (1/a^2)√(a^2 - x^2)/(-x) = -(1/a^2)√(a^2 - x^2)/x.
Those are the same, so the antiderivative is just -(1/a^2)√(a^2 - x^2)/x.
In your case a = 4, so the antiderivative is -(1/16)√(16 - x^2)/x.