Integral of sin(x) with Proof

The integral of sin(x) with respect to x is given by:

∫sin(x) dx = -cos(x) + C

where C is the constant of integration.

Proof:

To prove that ∫sin(x) dx = -cos(x) + C, we need to show that the derivative of -cos(x) is sin(x).

The derivative of cos(x) is -sin(x), so by the chain rule, the derivative of -cos(x) is sin(x).

d(-cos(x))/dx = sin(x)

This confirms that -cos(x) is the antiderivative of sin(x), hence ∫sin(x) dx = -cos(x) + C, where C is the constant of integration.