From Cosine to Sine

Using differential under integral sign we get the Ans=0+cos(x) (1/(cos(x))- (-sin(x)) (1/sin(x))=1+1=2

The integral of $\frac{1}{\sqrt{1-x^2}}$ is $\arcsin{x}$ . Recall that $\cos{x} = \sin(\pi/2 - x)$ , so if we we evaluate the integral from $\cos{x} \to \sin{x}$ We get: $\arcsin{\sin{x}} - \arcsin{\sin(pi/2+x)} = x - (pi/2 - x) = 2x - pi/2$ Taking the derivative we get the answer is 2.

I = $\int \frac{1}{1 - x^2}$ = arcsin x

So the result is

$\frac{d}{dx}$ [arcsin(sinx) - arcsin(cosx)]

Deriving using the Chain Rule the result is:

$\frac{cosx}{1- (cosx)^2} - \frac{-sinx}{1- (sinx)^2}$ = 2