The integral \( \displaystyle \int \sin x \, dx \) is a basic result in calculus, and is used in various calculations involving trigonometry, wave motion, and oscillatory behavior.

Solution

The antiderivative of \( \sin x \) with respect to \( x \) is:

\[\begin{align*} \displaystyle \int \sin x \, dx = -\cos x + C, \end{align*}\]

where \( C \) is the constant of integration.

Explanation

The derivative of \( -\cos x \) is \( \sin x \):
\[\begin{align*} \dfrac{d}{dx} -\cos x  = \sin x \end{align*}\]
To reverse this process and find the integral, we integrate both sides. Thus integral \( \sin x \) yields a negative cosine function, reflecting the cyclical relationship between these trigonometric functions.

Applications

Integrating \( \sin x \) is useful in many contexts, such as in:

• Physics: For modeling oscillations in springs, pendulums, and waves.
• Electronics: For analyzing AC circuits, where sine waves represent current or voltage.
• Signal Processing: For breaking down signals in Fourier analysis.

Practice Problem with Solution

Problem: A machine part oscillates with velocity \( v(t) = 4 \sin(t) \) m/s. Find the position function \( s(t) \) if \( s(0) = 2 \).

Solution:

\[\begin{align*} s(t) = \displaystyle \int 4 \sin(t) \, dt = -4 \cos(t) + C. \end{align*}\]

Using \( s(0) = 2 \):

\[\begin{align*} -4 \cos(0) + C = 2 \Rightarrow -4 + C = 2 \Rightarrow C = 6. \end{align*}\]

So, \( s(t) = -4 \cos(t) + 6 \).

Additional Problems with Solutions

• Evaluate  \( \displaystyle \int 3 \sin(x) \, dx \).
  \[\begin{align*} \displaystyle \int 3 \sin(x) \, dx = -3 \cos(x) + C. \end{align*}\]
• Compute  \( \displaystyle \int_0^{\pi} \sin(x) \, dx \).
  \[\begin{align*} \displaystyle \int_0^{\pi} \sin(x) \, dx =-\cos(x) \bigg|_0^{\pi} = -\cos(\pi) + \cos(0) = 1 + 1 = 2. \end{align*}\]
• Find  \( \displaystyle \int \sin(2x) \, dx \).
  \[\begin{align*} \displaystyle \int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C. \end{align*}\]