sin x 的積分：正弦的原函數

Problem

Find the integral of $\sin x$.

Step 1: Start with the meaning

Integration means finding an antiderivative, which is a function whose derivative gives the function we started with. For

$$\int \sin x\,dx,$$

the goal is to find a function whose derivative is $\sin x$.

Step 2: Use a known derivative

A key fact from trigonometric differentiation is

$$\frac{d}{dx}(\cos x)=-\sin x.$$

So $\cos x$ is close, but its derivative has the opposite sign.

Step 3: Adjust the sign

Since $\cos x$ differentiates to $-\sin x$, multiplying by $-1$ reverses the sign:

$$\frac{d}{dx}(-\cos x)=\sin x.$$

Therefore, $-\cos x$ is an antiderivative of $\sin x$.

Step 4: Add the constant

Every function that differs from $-\cos x$ by a constant has the same derivative, because the derivative of a constant is $0$. Therefore,

$$\int \sin x\,dx=-\cos x+C.$$